+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/preamble.ma".
-
-definition blt:
- nat \to (nat \to bool)
-\def
- let rec blt (m: nat) (n: nat) on n: bool \def (match n with [O \Rightarrow
-false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S m0)
-\Rightarrow (blt m0 n0)])]) in blt.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/blt/defs.ma".
-
-theorem lt_blt:
- \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
-(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0
-\def (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat
-n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq
-nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in
-nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) |
-(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind
-nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop)
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind
-((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal
-nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to
-(eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
-nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S
-n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n))
-\to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m
-n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1)))))
-y)))) x).
-
-theorem le_bge:
- \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
-(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O
-y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y:
-nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind
-(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false)))
-(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le return (\lambda (n0:
-nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n))
-false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
-(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
-(False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow
-(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e:
-nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
-| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool
-(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0:
-nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n))
-false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y))))
-x).
-
-theorem blt_lt:
- \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
-y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O)
-true)).(let H0 \def (match H in eq return (\lambda (b: bool).(\lambda (_: (eq
-? ? b)).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow
-(\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O)
-(\lambda (e: bool).(match e in bool return (\lambda (_: bool).Prop) with
-[true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind
-(lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda
-(H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda
-(y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt
-n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S
-O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq
-bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)
-\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H
-n0 H1))))) y)))) x).
-
-theorem bge_le:
- \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
-y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O)
-false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq
-bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda
-(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0:
-(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq return (\lambda
-(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O))))
-with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n))
-false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e
-in bool return (\lambda (_: bool).Prop) with [true \Rightarrow True | false
-\Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1
-(refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0
-(S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n))
-false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0
-H1))))))) y)))) x).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/types/defs.ma".
-
-include "Base-1/blt/defs.ma".
-
-include "Base-1/plist/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/preamble.ma".
-
-theorem nat_dec:
- \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to
-(\forall (P: Prop).P))))
-\def
- \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat
-n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2:
-nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall
-(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P:
-Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n)
-((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq
-nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S
-n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match
-ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
-\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n:
-nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to
-(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or
-(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror
-(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0:
-(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda
-(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda
-(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall
-(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
-Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P:
-Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0
-(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P:
-Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S
-n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat
-(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat
-(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P:
-Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to
-(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P:
-Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
-(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n)
-(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3)
-\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0
-(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0:
-Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2))))
-n1).
-
-theorem simpl_plus_r:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n)
-(plus p n)) \to (eq nat m p))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
-(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda
-(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
-nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n
-p)) (plus m n) H) (plus n m) (plus_sym n m)))))).
-
-theorem minus_Sx_Sy:
- \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
-\def
- \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
-
-theorem minus_plus_r:
- \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
-\def
- \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
-nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
-
-theorem plus_permute_2_in_3:
- \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
-y) z) (plus (plus x z) y))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x
-(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat
-(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind
-nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
-(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z
-y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
-
-theorem plus_permute_2_in_3_assoc:
- \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
-h) k) (plus n (plus k h)))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus
-(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r
-nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
-(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k
-h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
-
-theorem plus_O:
- \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
-x O) (eq nat y O))))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus
-n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda
-(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O)
-H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O)
-\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq
-nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0:
-nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq
-nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n)
-y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in
-nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y
-O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
-
-theorem minus_Sx_SO:
- \forall (x: nat).(eq nat (minus (S x) (S O)) x)
-\def
- \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal
-nat x) (minus x O) (minus_n_O x)).
-
-theorem eq_nat_dec:
- \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq
-nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or
-(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O)
-(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq
-nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j))
-(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq
-nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat
-(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S
-n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or
-(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq
-nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda
-(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S
-n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
-(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H
-n0)))) j)))) i).
-
-theorem neq_eq_e:
- \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j))
-\to P)) \to ((((eq nat i j) \to P)) \to P))))
-\def
- \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not
-(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def
-(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
-
-theorem le_false:
- \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S
-n) m) \to P))))
-\def
- \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P:
-Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P:
-Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match
-H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to
-P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
-(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
-(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0)
-O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
-I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal
-nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P:
-Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind
-(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n))
-\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S
-O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda
-(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2:
-(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e
-in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow
-(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda
-(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P)
-H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_:
-((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda
-(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S
-n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
-
-theorem le_Sx_x:
- \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
-\def
- \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
-le_Sn_n in (False_ind P (H0 x H))))).
-
-theorem le_n_pred:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
-(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
-nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
-(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
-
-theorem minus_le:
- \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n
-y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall
-(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0:
-nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda
-(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)])
-(S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
-
-theorem le_plus_minus_sym:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n)
-n))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
-(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
-(plus (minus m n) n) (plus_sym (minus m n) n)))).
-
-theorem le_minus_minus:
- \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
-\to (le (minus y x) (minus z x))))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z:
-nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x)
-(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
-z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
-(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
-
-theorem le_minus_plus:
- \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat
-(minus (plus x y) z) (plus (minus x z) y)))))
-\def
- \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to
-(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y))))))
-(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return
-(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y:
-nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n
-\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n:
-nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))
-(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O)
-(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq
-nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y:
-nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O
-m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))])
-in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x:
-nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus
-(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S
-z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
-(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def
-(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
-\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with
-[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat
-(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with
-[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq
-nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1)
-\Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m)
-(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S
-z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3))
-H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S
-z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
-(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n
-(le_S_n z0 n H1) y))))) x)))) z).
-
-theorem le_minus:
- \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to
-(le x (minus z y)))))
-\def
- \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus
-x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z
-y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x
-y))))).
-
-theorem le_trans_plus_r:
- \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to
-(le y z))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus
-x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
-
-theorem lt_x_O:
- \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
-\def
- \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
-(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
-ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
-\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))).
-
-theorem le_gen_S:
- \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
-nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
-\def
- \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def
-(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x)
-\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m
-n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
-(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0)))
-(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S
-m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x
-H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat
-(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq
-nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m)
-m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
-nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2))))
-x H1 H0))]) in (H0 (refl_equal nat x))))).
-
-theorem lt_x_plus_x_Sy:
- \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
-nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
-(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
-
-theorem simpl_lt_plus_r:
- \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
-p)) \to (lt n m))))
-\def
- \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus
-n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p)
-(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let
-H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
-(plus p m) (plus_sym m p)) in H1)))))).
-
-theorem minus_x_Sy:
- \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
-(minus x (S y))))))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
-(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H:
-(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_:
-(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with
-[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S
-y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
-(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda
-(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e:
-nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
-| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat
-(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O)))))
-(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat
-(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
-nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))
-(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S
-n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0:
-nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S
-n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0)
-n H1) in (H n0 H2))))) y)))) x).
-
-theorem lt_plus_minus:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus
-y (S x)))))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S
-x) y H))).
-
-theorem lt_plus_minus_r:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y
-(S x)) x)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
-(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
-y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
-
-theorem minus_x_SO:
- \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
-\def
- \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n:
-nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal
-nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
-
-theorem le_x_pred_y:
- \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
-\def
- \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to
-(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def
-(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
-\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let
-H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda
-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0)
-in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat
-(S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
-(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt
-x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S
-n))).(le_S_n x n H0))))) y).
-
-theorem lt_le_minus:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
-O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
-(plus_sym x (S O)))))).
-
-theorem lt_le_e:
- \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
-\to ((((le d n) \to P)) \to P))))
-\def
- \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n
-d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in
-(or_ind (le d n) (lt n d) P H0 H H1)))))).
-
-theorem lt_eq_e:
- \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
-\to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
-y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x
-y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
-
-theorem lt_eq_gt_e:
- \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
-\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
-y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
-\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda
-(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
-
-theorem lt_gen_xS:
- \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2
-nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S
-n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m)))
-(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S
-n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m)))
-(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda
-(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda
-(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0:
-nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat
-(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0)))
-(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt
-m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
-
-theorem le_lt_false:
- \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P:
-Prop).P))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt
-y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
-
-theorem lt_neq:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
-nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
-(lt_n_n y H1))))).
-
-theorem arith0:
- \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
-\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
-\def
- \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le
-(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1))
-h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2
-(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat
-(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat
-(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus
-(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1)
-(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2
-d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1)
-(minus_plus h2 (plus d2 h1))))))).
-
-theorem O_minus:
- \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
-(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O
-y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y:
-nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind
-(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
-x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0)
-O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0
-n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S
-x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O)
-H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n)
-\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0
-l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y))))
-x).
-
-theorem minus_minus:
- \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y)
-\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
-\def
- \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y:
-nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq
-nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O
-x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y
-O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n
-(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O)
-(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda
-(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to
-((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x
-y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le
-(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0)))
-\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda
-(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S
-z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le
-z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S
-x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4)))))
-(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y:
-nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus
-y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n:
-nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S
-z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S
-x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0))
-(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda
-(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O)
-(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0
-x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0))))))
-(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to
-((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0)
-y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S
-y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S
-z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0)
-H1))))))) y)))) x)))) z).
-
-theorem plus_plus:
- \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
-nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z
-x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
-\def
- \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2:
-nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq
-nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2)
-(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1:
-nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2
-O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat
-(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O
-(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def
-(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus
-O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1))))))))
-(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2:
-nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to
-((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus
-x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n:
-nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0))
-\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S
-z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2:
-nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
-(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus
-(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda
-(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O
-(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y
-\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
-nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq
-nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0))
-in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2)
-H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2:
-nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus
-(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2))
-\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
-nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda
-(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O
-x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
-nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S
-y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0
-(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n:
-nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus
-(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1))
-(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda
-(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus
-z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1))
-(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1))))))))))))
-x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1:
-nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat
-(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2
-y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n:
-nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S
-z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))
-\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda
-(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S
-z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let
-H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O)
-(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2)
-y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O
-z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2
-z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus
-x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def
-(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0)
-\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1)))))
-H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0)
-H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall
-(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus
-z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow
-(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4
-y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S
-z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0
-x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
-y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3))))
-x1)))) z).
-
-theorem le_S_minus:
- \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
-(le d (S (minus n h))))))
-\def
- \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
-d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
-\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
-(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S
-d (minus n h) (le_minus d n h H))))))).
-
-theorem lt_x_pred_y:
- \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
-\def
- \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
-n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
-(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
-(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x)
-n H0)))))) y)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/preamble.ma".
-
-theorem insert_eq:
- \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G:
-((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) \to
-((P x) \to (G x))))))
-\def
- \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G:
-((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to (G
-y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))).
-
-theorem unintro:
- \forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x:
-A).(P x))) \to (P a))))
-\def
- \lambda (A: Set).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda (H:
-((\forall (x: A).(P x)))).(H a)))).
-
-theorem xinduction:
- \forall (A: Set).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall (x:
-A).((eq A t x) \to (P x)))) \to (P t))))
-\def
- \lambda (A: Set).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda (H:
-((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/preamble.ma".
-
-inductive PList: Set \def
-| PNil: PList
-| PCons: nat \to (nat \to (PList \to PList)).
-
-definition PConsTail:
- PList \to (nat \to (nat \to PList))
-\def
- let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def
-(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow
-(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0
-h0 d0))]))) in PConsTail.
-
-definition Ss:
- PList \to PList
-\def
- let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow
-PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss.
-
-definition papp:
- PList \to (PList \to PList)
-\def
- let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b:
-PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons
-h d (papp a0 b))])) in papp.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/plist/defs.ma".
-
-theorem papp_ss:
- \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss
-is2)) (Ss (papp is1 is2))))
-\def
- \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2:
-PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2:
-PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0:
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp
-(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList
-(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n
-(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p
-is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Legacy-1/theory.ma".
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/theory.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/ext/tactics.ma".
-
-include "Base-1/ext/arith.ma".
-
-include "Base-1/types/props.ma".
-
-include "Base-1/blt/props.ma".
-
-include "Base-1/plist/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/preamble.ma".
-
-inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
-| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))).
-
-inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
-| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))).
-
-inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop
-\def
-| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3
-P4))))).
-
-inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
-| or3_intro0: P0 \to (or3 P0 P1 P2)
-| or3_intro1: P1 \to (or3 P0 P1 P2)
-| or3_intro2: P2 \to (or3 P0 P1 P2).
-
-inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
-| or4_intro0: P0 \to (or4 P0 P1 P2 P3)
-| or4_intro1: P1 \to (or4 P0 P1 P2 P3)
-| or4_intro2: P2 \to (or4 P0 P1 P2 P3)
-| or4_intro3: P3 \to (or4 P0 P1 P2 P3).
-
-inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop
-\def
-| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4).
-
-inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to
-Prop): Prop \def
-| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0
-P0 P1 P2)))).
-
-inductive ex4 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop)
-(P3: A0 \to Prop): Prop \def
-| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0)
-\to (ex4 A0 P0 P1 P2 P3))))).
-
-inductive ex_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)): Prop \def
-| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1
-P0))).
-
-inductive ex2_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to
-(A1 \to Prop)): Prop \def
-| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1)
-\to (ex2_2 A0 A1 P0 P1)))).
-
-inductive ex3_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to
-(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def
-| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1)
-\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))).
-
-inductive ex4_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to
-(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): Prop
-\def
-| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1)
-\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))).
-
-inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
-Prop))): Prop \def
-| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1
-x2) \to (ex_3 A0 A1 A2 P0)))).
-
-inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
-Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def
-| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
-x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))).
-
-inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
-Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to
-Prop))): Prop \def
-| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
-x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1
-P2)))))).
-
-inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
-Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to
-Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def
-| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
-x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0
-A1 A2 P0 P1 P2 P3))))))).
-
-inductive ex5_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
-Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to
-Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to
-Prop))): Prop \def
-| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
-x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0
-x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))).
-
-inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to
-(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0
-\to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def
-| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
-(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to
-(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))).
-
-inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to
-(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0
-\to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 \to (A3 \to
-Prop)))): Prop \def
-| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
-(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to
-((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))).
-
-inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to
-(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to
-(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3:
-A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def
-| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
-(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to
-((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1
-P2 P3))))))))).
-
-inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to
-(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to
-(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3:
-A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: A0 \to (A1 \to (A2 \to
-(A3 \to (A4 \to Prop))))): Prop \def
-| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
-(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to
-((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to
-(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))).
-
-inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set)
-(P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1: A0 \to
-(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P2: A0 \to (A1 \to (A2
-\to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to
-(A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5
-\to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to
-Prop)))))): Prop \def
-| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
-(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1
-x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5)
-\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2
-A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))).
-
-inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set)
-(A6: Set) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
-Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
-Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
-Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
-Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
-Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
-Prop))))))): Prop \def
-| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
-(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2
-x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6)
-\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1
-x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4
-P5))))))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-1/types/defs.ma".
-
-theorem ex2_sym:
- \forall (A: Set).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to
-Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A
-(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))))))
-\def
- \lambda (A: Set).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to
-Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q
-x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A
-(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0:
-(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda
-(x0: A).(P x0)) x H1 H0)))) H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/preamble.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/blt/defs.ma".
-
-inline procedural "Base-1/blt/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/preamble.ma".
-
-inline procedural "Base-1/ext/arith.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/preamble.ma".
-
-inline procedural "Base-1/ext/tactics.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/preamble.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/plist/defs.ma".
-
-inline procedural "Base-1/plist/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Base-1/definitions.ma".
-include "Legacy-2/theory.ma".
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/ext/tactics.ma".
-
-include "Base-2/ext/arith.ma".
-
-include "Base-2/types/props.ma".
-
-include "Base-2/blt/props.ma".
-
-include "Base-2/plist/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/preamble.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Base-2/types/defs.ma".
-
-inline procedural "Base-1/types/props.ma".
-
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/preamble.ma".
+
+inductive A: Set \def
+| ASort: nat \to (nat \to A)
+| AHead: A \to (A \to A).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+inductive C: Set \def
+| CSort: nat \to C
+| CHead: C \to (K \to (T \to C)).
+
+definition cweight:
+ C \to nat
+\def
+ let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O
+| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight.
+
+definition clt:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))).
+
+definition cle:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))).
+
+definition CTail:
+ K \to (T \to (C \to C))
+\def
+ let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n)
+\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k
+t d) h u)]) in CTail.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+include "LambdaDelta-1/T/props.ma".
+
+theorem clt_cong:
+ \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
+T).(clt (CHead c k t) (CHead d k t))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
+d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
+(tweight t) H))))).
+
+theorem clt_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
+c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u))))
+(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
+(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
+
+theorem clt_wf__q_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to
+Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0
+c))))) P n))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
+n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight
+c)))))).
+
+theorem clt_wf_ind:
+ \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
+\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
+(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind
+(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0:
+C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
+(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
+(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P
+c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt
+(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight
+d))))))))))))) c)))).
+
+theorem chead_ctail:
+ \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
+K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t)
+(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k:
+K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead
+(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall
+(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def
+H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d:
+C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d))))))
+(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead
+c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1:
+C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead
+c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d:
+C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
+(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0
+k t) H1))))) H0))))))))) c).
+
+theorem clt_thead:
+ \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt
+c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
+C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t:
+T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
+
+theorem c_tail_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
+(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t
+c))))))) \to (\forall (c: C).(P c))))
+\def
+ \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort
+n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t:
+T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P
+c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1)
+\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d:
+C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_:
+((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to
+(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t)
+(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2
+x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def
+(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P
+d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2))
+(CHead c1 k t) H4))))) H3)))))))) c0)) c)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/preamble.ma".
+
+record G : Set \def {
+ next: (nat \to nat);
+ next_lt: (\forall (n: nat).(lt n (next n)))
+}.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+theorem terms_props__bind_dec:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq
+B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b:
+B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl
+(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B
+Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P:
+Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def
+(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow
+False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B
+Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda
+(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda
+(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall
+(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall
+(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def
+(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
+False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B
+Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B
+Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
+Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P
+H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B
+Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void
+Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P:
+Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
+Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B
+Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
+Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P
+H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P:
+Prop).P)) (refl_equal B Void)) b2)) b1).
+
+theorem bind_dec_not:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
+\def
+ \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2)
+in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P:
+Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
+b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
+(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1
+b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
+
+theorem terms_props__flat_dec:
+ \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq
+F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f:
+F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl
+(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F
+Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P:
+Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def
+(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop)
+with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in
+(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F
+Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast
+Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast
+Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match
+ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast
+\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast
+Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2))
+f1).
+
+theorem terms_props__kind_dec:
+ \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq
+K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind
+(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P:
+Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in
+(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P:
+Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to
+(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1:
+B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P:
+Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to
+(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B
+b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq
+K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b)
+(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e:
+K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
+(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B
+b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in
+(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b)
+(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H:
+(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b)
+(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind
+P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or
+(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda
+(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to
+(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P:
+Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def
+(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F
+f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat
+f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind
+F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1))
+\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat
+f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0))
+(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K
+(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P))
+(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def
+(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with
+[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1)
+in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall
+(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1).
+
+theorem term_dec:
+ \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq
+T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2:
+T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to
+(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in
+(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
+Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to
+(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda
+(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to
+(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort
+n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0))
+(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T
+(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P))
+(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _)
+\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0
+(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in
+(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T
+(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P))
+(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0))))))
+(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T
+(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or
+(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P:
+Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n)
+(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n)
+(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n:
+nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T
+(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T
+(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P))
+(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0))))))
+(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind
+(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n)
+(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda
+(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n)
+(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P))))
+(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to
+(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq
+nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0))
+((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T
+(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n]))
+(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1:
+nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal
+nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T
+(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0:
+T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P:
+Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n)
+(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n)
+(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t
+t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall
+(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P:
+Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t
+t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort
+n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort
+n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T
+(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P:
+Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P:
+Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in
+(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq
+T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P:
+Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq
+T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in
+(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P:
+Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0)
+(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let
+H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
+(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t
+(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t
+t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in
+(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P:
+Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0)
+(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let
+H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
+(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0
+(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t
+t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def
+(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq
+K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0))
+((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda
+(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0)
+(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P:
+Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t
+t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0)))
+k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror
+(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0))
+\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t
+t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
+t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to
+(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9)))
+t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror
+(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4))
+\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t
+t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
+t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in
+(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq
+T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r
+T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5)
+\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P)))))
+H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P:
+Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k
+t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead
+k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
+(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t
+| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0)
+(THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3
+t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def
+(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k
+t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r
+T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in
+(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5)
+((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12
+(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
+
+theorem binder_dec:
+ \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
+T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall
+(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b:
+B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w
+u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind
+b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n)
+(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
+(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w
+u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n:
+nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
+T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P:
+Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq
+T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T
+(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0)))))))))
+(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
+u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead
+(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3
+B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind
+b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
+(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda
+(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b)
+w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0
+t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b:
+B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda
+(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0:
+B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to
+(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T
+(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w
+u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
+(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T
+(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1)
+(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u:
+T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P:
+Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u:
+T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal
+T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda
+(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0
+(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u:
+T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda
+(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda
+(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P:
+Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w:
+T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))))))
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1)
+(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
+(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead
+(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0
+t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
+\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1)
+in (False_ind P H2))))))))))))) k)) t).
+
+theorem abst_dec:
+ \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead
+(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to
+(\forall (P: Prop).P)))))
+\def
+ \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda
+(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead
+(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda
+(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v
+t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall
+(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind
+Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n:
+nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead
+(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t))
+\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n)
+(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0:
+T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind
+Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_:
+((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v
+t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P:
+Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind
+Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind
+Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t
+t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead
+(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind
+Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1:
+T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T
+(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let
+H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t
+v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind
+Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind
+Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda
+(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq
+T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq
+T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P:
+Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0)
+(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0)
+(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda
+(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0
+(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v)
+\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead
+(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead
+(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))
+(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind
+Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t |
+(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst)
+t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind
+Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8
+P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall
+(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead
+(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind
+Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T
+(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
+\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _)
+\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
+\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_:
+(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4)))))))
+H1))))))))) u).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/preamble.ma".
+
+inductive B: Set \def
+| Abbr: B
+| Abst: B
+| Void: B.
+
+inductive F: Set \def
+| Appl: F
+| Cast: F.
+
+inductive K: Set \def
+| Bind: B \to K
+| Flat: F \to K.
+
+inductive T: Set \def
+| TSort: nat \to T
+| TLRef: nat \to T
+| THead: K \to (T \to (T \to T)).
+
+definition tweight:
+ T \to nat
+\def
+ let rec tweight (t: T) on t: nat \def (match t with [(TSort _) \Rightarrow
+(S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus
+(tweight u) (tweight t0)))]) in tweight.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+theorem not_abbr_abst:
+ not (eq B Abbr Abst)
+\def
+ \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True |
+Abst \Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind
+False H0)).
+
+theorem not_void_abst:
+ not (eq B Void Abst)
+\def
+ \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind
+False H0)).
+
+theorem not_abbr_void:
+ not (eq B Abbr Void)
+\def
+ \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True |
+Abst \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind
+False H0)).
+
+theorem not_abst_void:
+ not (eq B Abst Void)
+\def
+ \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind
+False H0)).
+
+theorem thead_x_y_y:
+ \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq
+T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda
+(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in
+(False_ind P H0))))) (\lambda (n: nat).(\lambda (H: (eq T (THead k v (TLRef
+n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (TLRef
+n)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0:
+K).(\lambda (t0: T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P:
+Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to
+(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1))
+(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e:
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
+(TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (THead
+k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v |
+(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) (THead k v (THead
+k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead
+k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2)
+\Rightarrow t2])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in
+(\lambda (H5: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T
+v (\lambda (t2: T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0)))
+H0 t0 H5) in (let H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0
+t1) t1) \to (\forall (P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3))
+H2))))))))) t))).
+
+theorem tweight_lt:
+ \forall (t: T).(lt O (tweight t))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_:
+nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda
+(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O
+(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S
+O) (tweight t0) (tweight t1) H))))))) t).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/asucc/defs.ma".
+
+definition aplus:
+ G \to (A \to (nat \to A))
+\def
+ let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O
+\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/aplus/defs.ma".
+
+include "LambdaDelta-1/next_plus/props.ma".
+
+theorem aplus_reg_r:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
+(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A
+(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h:
+nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2
+(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n
+h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n
+h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))).
+
+theorem aplus_assoc:
+ \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A
+(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n:
+nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n
+h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n:
+nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus
+g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A
+(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))))
+(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g
+(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O
+n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n))
+n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda
+(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g
+(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n))
+n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0))
+(plus_n_Sm n n0)))) h2)))) h1))).
+
+theorem aplus_asucc:
+ \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a)
+h) (asucc g (aplus g a h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a
+(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h))))
+(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h)
+(aplus_assoc g a (S O) h)))).
+
+theorem aplus_sort_O_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O
+n) (S k)) (aplus g (ASort O (next g n)) k))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc
+g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k)))
+(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n)
+k)) (aplus_asucc g k (ASort O n))))).
+
+theorem aplus_sort_S_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A
+(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind
+A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g
+(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g
+(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
+
+theorem aplus_asort_O_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O
+n) h) (ASort O (next_plus g n h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0:
+nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda
+(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall
+(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0
+n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n)
+(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat
+(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next
+g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n))
+(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n
+(ASort O n0)))))) h)).
+
+theorem aplus_asort_le_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h
+k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k:
+nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort
+(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O
+k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n)))
+(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0:
+nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A
+(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k:
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A
+(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda
+(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat
+O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n)
+h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S
+x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
+| (S _) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus
+g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0))))
+(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A
+(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda
+(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort
+(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n
+n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g
+h0 (ASort (S n) n0))))))) k)))) h)).
+
+theorem aplus_asort_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A
+(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n:
+nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus
+g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k))
+(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h)
+(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k)
+(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus
+h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a
+(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O
+(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k
+h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A
+(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k)))))
+(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h
+(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k)
+(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus
+h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h
+(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort
+(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n
+(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h)
+n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h)
+(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h)
+(aplus_asort_le_simpl g h k n H))))))).
+
+theorem aplus_ahead_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A
+(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1:
+A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2
+n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2))))
+(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A
+(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1:
+A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda
+(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g
+(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n)
+(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n
+a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2)))))))
+h)).
+
+theorem aplus_asucc_false:
+ \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a)
+h) a) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h:
+nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A
+(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0)
+\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind
+(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O
+(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P))
+(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1
+\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0
+(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
+(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat
+(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1)
+\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1:
+nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0)
+\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0)
+(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
+(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1:
+nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in
+(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g
+n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n
+(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda
+(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort
+O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to
+P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let
+H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort
+(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
+(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow
+n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat)
+\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow
+(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in
+minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S
+n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _)
+\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def
+(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0
+n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus
+g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1
+h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2
+n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0:
+A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to
+(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h:
+nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P:
+Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc
+g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g
+(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1
+(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1)))
+in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n:
+nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc
+g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow
+a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3
+P)))))))))) a)).
+
+theorem aplus_inj:
+ \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A
+(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2:
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
+h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A
+(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_:
+(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a:
+A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0:
+(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g
+a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g
+n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq
+nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2:
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
+h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq
+A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a:
+A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A
+(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a)
+n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O)))))
+(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a
+n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1:
+(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def
+(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus
+g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def
+(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g
+a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat
+nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+definition cbk:
+ C \to nat
+\def
+ let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m |
+(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk.
+
+definition app1:
+ C \to (T \to T)
+\def
+ let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with
+[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u
+t))])) in app1.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/A/defs.ma".
+
+inductive aprem: nat \to (A \to (A \to Prop)) \def
+| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
+| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2
+a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/aprem/defs.ma".
+
+theorem aprem_gen_sort:
+ \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem
+i (ASort h n) x) \to False))))
+\def
+ \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n:
+nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda
+(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0:
+(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_:
+A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2:
+A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A
+(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
+(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a:
+A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2
+(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2)
+(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee
+in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False
+H4))))))))) i y x H0))) H))))).
+
+theorem aprem_gen_head_O:
+ \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x)
+\to (eq A x a1))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O
+(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x))
+(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y
+x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq
+A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem
+y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq
+nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0:
+A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0
+a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _)
+\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3)
+in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a:
+A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i
+O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda
+(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let
+H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3
+a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e
+in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _
+a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A
+a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A
+a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0
+(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i)
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a
+a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
+
+theorem aprem_gen_head_S:
+ \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem
+(S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda
+(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a:
+A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y:
+A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n:
+nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2
+x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n:
+nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead
+a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda
+(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let
+H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3)
+(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a)
+\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0
+a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat
+O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind
+(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a:
+A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat
+i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3:
+A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0)
+(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _)
+\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2)
+H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4:
+A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3
+a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2
+H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat
+return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n]))
+(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n
+(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let
+H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in
+H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/aprem/fwd.ma".
+
+include "LambdaDelta-1/leq/defs.ma".
+
+theorem aprem_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
+(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g
+b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall
+(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
+(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda
+(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g
+(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2:
+A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2
+i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g
+b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda
+(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall
+(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g
+b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda
+(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall
+(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
+(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda
+(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead
+a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem
+n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y
+\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A
+(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))
+(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead
+a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda
+(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2))
+(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0)
+(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let
+H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1:
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1:
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda
+(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2
+A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0
+a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2
+H)))).
+
+theorem aprem_asucc:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i
+a1 a2) \to (aprem i (asucc g a1) a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda
+(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda
+(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3:
+A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda
+(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0)
+a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2
+H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/arity/props.ma".
+
+include "LambdaDelta-1/arity/cimp.ma".
+
+include "LambdaDelta-1/aprem/props.ma".
+
+theorem arity_aprem:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0:
+A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda
+(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b
+i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_:
+(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem
+i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b:
+A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def
+H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop
+(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0
+\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda
+(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
+(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop
+(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus
+i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2)
+H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2:
+((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem
+i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6:
+(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2)
+x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1:
+C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1
+x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop
+(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x
+x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8
+(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7))))))))
+H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d
+u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i:
+nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b)
+u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5:
+(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind
+C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O
+d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0
+(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9
+\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S
+b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g
+a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1)
+b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3:
+((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead
+c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b:
+A).(\lambda (H4: (aprem i (AHead a1 a2) b)).(nat_ind (\lambda (n:
+nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H5:
+(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in
+(eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a0))))))) (ex2_3_intro C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g a1))))) c0 u O (drop_refl c0) H0) b H_y))) (\lambda (i0:
+nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def
+(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6
+\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d:
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j)
+O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst)
+u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i
+H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b:
+A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3:
+((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem
+i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j))
+O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda
+(H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i
+x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))))
+(\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n)
+c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0
+(CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))
+(\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2))
+O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2))
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8)))))
+(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g
+d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k:
+K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1)
+c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda
+(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i
+x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10:
+(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r
+(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))
+(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n:
+nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d
+c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda
+(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r
+(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g
+(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in
+(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0:
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop
+(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g
+b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9
+(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7))))))
+H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda
+(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b:
+A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0:
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall
+(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i:
+nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4)
+in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6:
+(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g
+b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5))))))))))))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
+t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d
+u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1
+a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x
+\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A
+(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
+g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i
+a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0
+x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g
+x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7))))))
+H4))))))))))))) c t a H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/arity/defs.ma".
+
+include "LambdaDelta-1/cimp/props.ma".
+
+theorem arity_cimp_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g
+c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d
+c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c
+c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i)
+a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let
+H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in
+(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7:
+(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def
+(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead
+x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind
+Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _)
+\Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono
+c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11
+\def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9
+x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x
+H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6)))))
+H4))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2:
+C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda
+(H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in
+(ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2
+(TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst)
+u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def
+H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0:
+C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst)
+u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i
+c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i
+H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x |
+(CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind
+Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u)
+H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0
+(Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0:
+C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12)))))))))
+H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b
+Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
+g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b)
+u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c
+c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u)
+(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
+((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to
+(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
+(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2:
+C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2:
+C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2
+H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
+(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to
+(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
+a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0
+a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1
+c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
+C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2:
+(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2
+t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/leq/defs.ma".
+
+include "LambdaDelta-1/getl/defs.ma".
+
+inductive arity (g: G): C \to (T \to (A \to Prop)) \def
+| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O
+n)))
+| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a)
+\to (arity g c (TLRef i) a)))))))
+| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u
+(asucc g a)) \to (arity g c (TLRef i) a)))))))
+| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c:
+C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t:
+T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c
+(THead (Bind b) u t) a2)))))))))
+| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
+(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind
+Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
+| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
+a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to
+(arity g c (THead (Flat Appl) u t) a2)))))))
+| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u
+(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat
+Cast) u t) a))))))
+| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t
+a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/arity/defs.ma".
+
+include "LambdaDelta-1/leq/asucc.ma".
+
+include "LambdaDelta-1/getl/drop.ma".
+
+theorem arity_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c
+(TSort n) a) \to (leq g a (ASort O n))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda
+(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g
+c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0:
+(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0:
+A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_:
+C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1:
+nat).(leq g (ASort O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2)))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(_: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity
+g d u a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O
+n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T
+(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n))
+H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n))
+\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort
+n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
+(False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_:
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq
+g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
+(CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2
+(ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7
+\def (eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in
+(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda
+(_: (((eq T u (TSort n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t
+a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda
+(H5: (eq T (THead (Bind Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead
+(Bind Abst) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort n) H5) in (False_ind (leq g (AHead a1 a2)
+(ASort O n)) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq
+g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
+c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TSort n)) \to (leq g (AHead a1
+a2) (ASort O n))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TSort
+n))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
+H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g
+a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O
+n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t
+(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat
+Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n)) H6)))))))))))
+(\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t
+a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O
+n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t
+(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in
+(let H6 \def (eq_ind T t (\lambda (t0: T).((eq T t0 (TSort n)) \to (leq g a1
+(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t0:
+T).(arity g c0 t0 a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1
+a2 H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0)))
+H))))).
+
+theorem arity_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c
+(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a))))
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda
+(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g
+c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl
+i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))
+(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0:
+C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0:
+C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (ASort O n)))))))
+H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0:
+nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (H2: (arity g d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr)
+u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g
+a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal
+T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort
+_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n:
+nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6 H2)))))))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda
+(H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2:
+(arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2
+C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g
+a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal
+T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort
+_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n:
+nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6
+H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
+C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
+(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead
+c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda
+(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind
+b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
+(a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u
+(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i))
+\to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind
+Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity
+g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0
+(Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst)
+u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda
+(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
+c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1
+a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let H6
+\def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
+(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
+d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2))))
+(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst)
+u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))
+H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
+(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2
+C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g
+a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_:
+(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
+T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+(asucc g a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef
+i))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i)
+H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
+Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))
+H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1:
+(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2:
+A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5
+\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind
+T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind
+T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6
+(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2))))))
+(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda
+(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11:
+(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))
+x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11:
+(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2)
+(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
+
+theorem arity_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c:
+C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind
+b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_:
+A).(arity g (CHead c (Bind b) u) t a2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda
+(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity
+g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
+T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u
+a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y:
+T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda
+(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
+a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n)
+(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind b) u t) H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3)))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity
+g d u0 a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t
+a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
+(Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
+u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_:
+(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u
+a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g
+a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
+(Bind b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0
+Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3:
+(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g
+(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u
+t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3))
+(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t
+a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u
+t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead
+k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead
+(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b0) u0 t0)
+(THead (Bind b) u t) H7) in ((let H10 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda (H12:
+(eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead
+(Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u
+a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t
+a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g
+(CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def (eq_ind T u0
+(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
+A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: A).(arity g (CHead
+(CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) in (let H16 \def
+(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) t1) t a0)) H14 u
+H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b)
+u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let H18 \def (eq_ind T u0
+(\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let H19 \def (eq_ind B b0
+(\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
+A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: A).(arity g (CHead
+(CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in (let H20 \def
+(eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t a0)) H16 b
+H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H2
+b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_:
+A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) H9))
+H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
+(H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 (THead (Bind b) u
+t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: T).(\lambda (a0:
+A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (H5:
+(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead
+c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind
+Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead (Bind Abst) u0
+t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e
+in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _)
+\Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0
+u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1:
+T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g
+(CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0
+(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0
+(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let
+H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda
+(_: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u
+H10) in (let H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind
+Abst) t1) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1:
+T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u
+a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u
+H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g
+a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b0: B).((eq T t
+(THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind
+Abst) u) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u)
+(Bind b0) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda
+(b0: B).((eq T u (THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1))))))
+H16 Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
+Abst))) H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t
+(AHead a1 a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) in False
+return (\lambda (_: False).(ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with
+[]) in H21) b H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq
+T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0:
+T).(\lambda (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_:
+(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u
+a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1
+a0))))))).(\lambda (H6: (eq T (THead (Flat Appl) u0 t0) (THead (Bind b) u
+t))).(let H7 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)))
+H7)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_:
+(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t))
+\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_:
+(arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b)
+u t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b)
+u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1
+a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T
+(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0
+(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
+a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda
+(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7
+(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11:
+(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10
+(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y
+a2 H1))) H0)))))))).
+
+theorem arity_gen_abst:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
+A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) u) t a2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
+A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead
+(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A
+A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2:
+A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0:
+(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
+A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n:
+nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H1) in
+(False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A (ASort O n)
+(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0:
+A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst)
+u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1
+a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda
+(_: A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
+(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
+i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g
+a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A
+(\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
+(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
+(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
+i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1:
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead
+a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
+(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
+a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0
+(Bind b) u0) t0 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b)
+u0) (Bind Abst) u) t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0)
+(THead (Bind Abst) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e
+in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
+(Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
+(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
+(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0
+u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1:
+T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
+A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t
+H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind
+b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T
+t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
+A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead
+c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let
+H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2))
+H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead
+(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
+a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0
+t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t
+(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
+A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0
+(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let
+H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2))
+H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
+Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in
+False return (\lambda (_: False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4:
+A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
+(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
+Abst) u) t a4))))) with []) in H21))))))))))))) H8)) H7))))))))))))))
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0
+u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2
+a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
+(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
+a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0
+(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc
+g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
+Abst) u0) (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst)
+u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
+(TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind
+Abst) u0 t0) (THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
+(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T
+u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead
+a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0)
+u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0
+(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T
+t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in
+(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t))
+\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc
+g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
+Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0
+(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let
+H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3
+a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
+(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))
+H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc
+g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A
+(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
+(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
+Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12)))))))))
+H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
+(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
+A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda
+(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T
+(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T
+(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
+A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
+a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0:
+A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
+(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A
+(asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u
+(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind
+Abst) u) t a2))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
+a0)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
+(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity
+g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Cast)
+u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t)
+H5) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0
+(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
+A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead
+a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
+(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
+a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T
+t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0
+(THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t1:
+T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
+A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u t) H5) in (let H7
+\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Bind Abst)
+u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind Abst) u t))) in
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A
+(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0: A).(\lambda
+(x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10: (arity g c0 u
+(asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u) t x1)).(let
+H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in
+(let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u
+t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12)
+in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq
+g x0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3:
+A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
+A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
+(CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda
+(H15: (leq g x0 x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2
+(AHead x2 x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2
+x3) H17) in (eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda
+(a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda
+(_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity
+g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3:
+A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead
+x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2
+H15)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18)))))))
+H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
+
+theorem arity_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2:
+A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity
+g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2:
+A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead
+(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A
+(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1
+a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda
+(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t))
+\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
+(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
+(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O
+n))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a:
+A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl)
+u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g
+d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u
+t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u
+t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
+A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
+(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g
+d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda
+(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef
+i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a))))
+H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
+a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
+(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
+a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
+(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) u a3)) (\lambda (a3:
+A).(arity g (CHead c0 (Bind b) u0) t (AHead a3 a0))))))).(\lambda (H6: (eq T
+(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T
+(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda
+(c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc
+g a1))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
+(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (asucc g
+a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
+(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda
+(a3: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a3 a0))))))).(\lambda
+(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def
+(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1
+a0))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
+A).(\lambda (H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat
+Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
+A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0:
+A).(\lambda (H3: (arity g c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0
+(THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))))).(\lambda (H5:
+(eq T (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t))).(let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
+\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in
+((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _
+t1) \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5)
+in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq
+T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let
+H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t
+H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat
+Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
+A).(arity g c0 t (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0
+(\lambda (t1: T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3:
+A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12
+H10))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a:
+A).(\lambda (_: (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead
+(Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda
+(a1: A).(arity g c0 t (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda
+(_: (arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
+(AHead a1 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat
+Appl) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
+\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
+False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H5) in
+(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
+g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T
+t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (a3: A).(arity g c0 t (AHead a3 a1))))))).(\lambda (a0: A).(\lambda
+(H3: (leq g a1 a0)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5
+\def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let
+H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t
+(AHead a3 a1)))))) H2 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T
+t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in
+(let H8 \def (H6 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
+a1))) (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0
+t (AHead a3 a0)))) (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda
+(H10: (arity g c0 t (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0
+u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t
+(AHead x a1) H10 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3))))))
+H8))))))))))))) c y a2 H0))) H)))))).
+
+theorem arity_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
+A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a))
+(arity g c t a)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
+A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead
+(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land
+(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0:
+(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
+A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0))
+(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
+(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u
+(asucc g (ASort O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
+a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u
+(asucc g a0)) (arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat
+Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
+t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
+H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0:
+A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
+(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t
+(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u
+t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
+t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
+H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
+a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u
+(asucc g a1)) (arity g c0 t a1))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T
+t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind b) u0) u
+(asucc g a2)) (arity g (CHead c0 (Bind b) u0) t a2))))).(\lambda (H6: (eq T
+(THead (Bind b) u0 t0) (THead (Flat Cast) u t))).(let H7 \def (eq_ind T
+(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g c0 t
+a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead
+(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a1))) (arity g c0
+t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g
+(CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast)
+u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g
+(CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0
+t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t)
+H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t
+(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead
+a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g
+c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5:
+(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def
+(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land
+(arity g c0 u (asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g
+a0))).(\lambda (H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0
+u (asucc g (asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0:
+T).(\lambda (H3: (arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t
+a0))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Cast) u
+t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t)
+H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t1) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat
+Cast) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0
+(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
+(asucc g a0)) (arity g c0 t a0)))) H4 t H7) in (let H10 \def (eq_ind T t0
+(\lambda (t1: T).(arity g c0 t1 a0)) H3 t H7) in (let H11 \def (eq_ind T u0
+(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
+(asucc g (asucc g a0))) (arity g c0 t (asucc g a0))))) H2 u H8) in (let H12
+\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a0))) H1 u H8) in
+(conj (arity g c0 u (asucc g a0)) (arity g c0 t a0) H12 H10)))))))
+H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u t))
+\to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (a2:
+A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u
+t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Cast) u t)
+H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2
+(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
+T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6
+(refl_equal T (THead (Flat Cast) u t))) in (land_ind (arity g c0 u (asucc g
+a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2))
+(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t
+a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u
+(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1
+H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))).
+
+theorem arity_gen_appls:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall
+(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a:
+A).(arity g c t a))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads
+(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda
+(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c
+t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall
+(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a:
+A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g
+c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g
+c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1
+a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_:
+(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x
+a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A
+(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a)))
+(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a:
+A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
+
+theorem arity_gen_lift:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h:
+nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2:
+C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T
+(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall
+(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda
+(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0))
+\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat
+d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2:
+C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c:
+C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq
+T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0:
+T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda
+(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0
+(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1)))))))))
+(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2:
+(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u
+(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i)
+(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
+i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
+(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
+t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
+a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in
+(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h
+(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h
+(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i)
+(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef
+(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda
+(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda
+(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h)
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0
+H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst)
+u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda
+(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
+(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda
+(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x
+x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
+i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
+(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
+t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
+(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def
+(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus
+x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1
+(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt
+Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7:
+(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x
+h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le
+(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T
+(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0
+u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5
+H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1:
+(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall
+(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
+(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4:
+(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x:
+nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h
+x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x:
+nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x
+x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda
+(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda
+(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u
+(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T
+(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def
+(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
+(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to
+(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T
+t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x)
+x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c
+(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def
+(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift
+h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind
+b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15
+\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T
+t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
+a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1:
+T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1
+(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal
+T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b
+x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6))))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u
+(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u
+(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g
+(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall
+(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c
+(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda
+(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda
+(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0:
+T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1
+x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S
+x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2
+t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3:
+nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h
+x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x)
+x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c
+(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u
+(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11
+(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3:
+nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall
+(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2)))))))
+H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall
+(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3:
+C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1)
+H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g
+a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T
+(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2))
+(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0
+H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda
+(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
+(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4:
+((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall
+(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda
+(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift
+h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T
+(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
+(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1
+x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x
+x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1
+a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
+(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def
+(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2)
+H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
+(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
+(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2
+x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2
+(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0
+x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
+A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x:
+nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x
+c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3:
+(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T
+t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead
+(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c
+c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0)
+(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast)
+x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h
+x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1
+a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
+(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0
+(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def
+(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
+(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
+(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
+(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in
+(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10
+x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast
+u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
+A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall
+(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
+(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
+a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x
+x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1
+(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/arity/props.ma".
+
+include "LambdaDelta-1/drop1/fwd.ma".
+
+theorem arity_lift1:
+ \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds:
+PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t
+a) \to (arity g c1 (lift1 hds t) a))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t:
+T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))
+(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda
+(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C
+c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1:
+C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1
+(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1
+(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def
+(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
+(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1
+(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1
+x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1)
+c1 n n0 H3)))) H2))))))))))) hds)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/arity.ma".
+
+include "LambdaDelta-1/pr3/defs.ma".
+
+include "LambdaDelta-1/pr1/defs.ma".
+
+include "LambdaDelta-1/wcpr0/getl.ma".
+
+include "LambdaDelta-1/pr0/fwd.ma".
+
+include "LambdaDelta-1/arity/subst0.ma".
+
+theorem arity_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
+c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1
+t2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to
+(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2:
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n)
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n)))
+(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d
+(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda
+(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to
+(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i)
+(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda
+(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2
+(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2
+(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u
+x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2
+H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g
+a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2:
+T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2:
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i)
+t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
+T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0
+d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1
+H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i
+H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u
+a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
+u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda
+(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2:
+C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq
+T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g
+c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0:
+T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2))))
+(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T
+(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0
+c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp
+c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1
+(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b)
+u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k
+u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e:
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
+(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
+(THead (Bind b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3)
+(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
+(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3)
+(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K
+k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2
+t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind
+b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3
+(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1
+(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9
+u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16)
+in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b)
+u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14))
+H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g
+c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
+t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4
+a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
+(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead
+(Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+b) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2)
+H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
+(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_:
+(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4
+a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3))
+(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead
+(Bind b0) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity
+g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda
+(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3)
+(THead (Bind b) u t) H13) in ((let H15 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind
+Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0]))
+(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T
+u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0:
+T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in
+(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let
+H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to
+(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0:
+T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0:
+B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in
+(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t))
+\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b
+(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to
+(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let
+H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2))
+H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
+Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w
+a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr)
+u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr
+c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda
+(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9:
+(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2
+t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S
+O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 |
+(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in
+((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0
+_) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u
+t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t5)
+\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in
+lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0)
+\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t)
+H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def
+(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18
+\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to
+(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T
+t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to
+(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O
+t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c
+(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2
+(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u
+(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12))))))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_:
+(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0:
+T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let
+H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2)
+H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g
+c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2:
+(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2:
+C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t)
+t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2))
+(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6:
+(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind
+Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda
+(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda
+(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u
+t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1
+c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u
+u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1
+(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
+(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k:
+K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
+\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _)
+\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0)
+\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda
+(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind
+Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17
+\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to
+(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3
+(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1
+(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead
+a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0
+u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead
+c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k
+H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t))
+\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t))
+\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl)
+v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind
+T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead
+(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2
+(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
+u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2
+(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
+t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4
+(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b)
+u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl)
+v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1
+(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead
+(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w:
+T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr)
+u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T (THead (Bind Abbr)
+u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g
+c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b:
+B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u
+t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq
+T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11
+\def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B)
+with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O
+t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
+(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b)
+u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T
+\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T
+\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0)
+\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u
+t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let
+H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0))
+\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18
+\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind
+Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3
+(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity
+g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def
+(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g
+c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
+(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0:
+T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u
+t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Abst) u t) H9) in (False_ind (arity g c2
+t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda
+(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to
+(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2:
+(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2)
+\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1
+a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2:
+T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat
+Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2))
+(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda
+(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda
+(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def
+(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T
+(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2
+u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8))))
+(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8:
+(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda
+(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal
+T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0]))
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0]))
+(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1
+u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda
+(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda
+(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t
+H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in
+(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u
+t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda
+(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4
+a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0:
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8:
+(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u
+t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead
+_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
+(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead
+(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead
+_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
+(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def
+(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g
+c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0
+v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3
+(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0
+t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead
+(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3)
+H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0
+c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1
+a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3)
+H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead
+(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in
+(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A
+(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2
+(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0
+(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let
+H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1
+a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match
+e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _
+a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq
+A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2
+(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda
+(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr
+not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4
+a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1
+H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b:
+B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u
+t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to
+(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0
+t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4
+a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead
+(Bind b) u1 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T
+v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat
+Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1
+(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t
+(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2)))
+H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0:
+T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead
+(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u
+(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3)
+H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0
+c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1
+a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16)
+in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind
+b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def
+(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda
+(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2)
+t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda
+(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7
+c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead
+c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2
+(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2
+H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2:
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t))
+\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4
+a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T
+(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T
+(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2)
+H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda
+(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u
+t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind
+(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda
+(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity
+g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3)
+(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t)
+H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u
+(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall
+(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t:
+T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda
+(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0
+(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda
+(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y:
+T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq
+T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0:
+T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T
+(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat
+Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2
+H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1
+(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat
+Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T
+(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda
+(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
+| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
+(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3)
+(THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
+(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3)
+(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16:
+(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2
+(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def
+(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind
+T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2
+a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2))
+H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4
+H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1:
+T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead
+(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t))
+\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead
+(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) u t) H11) in (False_ind (arity
+g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\lambda (b: B).(\lambda
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2
+a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T
+(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let
+H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t)
+H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2:
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t))
+\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4
+a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T
+(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \def (eq_ind T
+(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0)
+H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda
+(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u
+t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H10) in (False_ind
+(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to
+(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast)
+u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
+(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat
+Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0]))
+(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T
+u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat
+Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3
+(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10))))))))
+y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1:
+A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda
+(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2
+H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))).
+
+theorem arity_sred_wcpr0_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall
+(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1
+c2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a:
+A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0
+a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a:
+A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1
+c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a:
+A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5
+a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3:
+(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a
+(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl
+c2)))))))))))))) t1 t2 H))).
+
+theorem arity_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g:
+G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda
+(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda
+(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0
+g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
+G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a
+(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t
+H2)))))))))))))) c t1 t2 H)))).
+
+theorem arity_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a:
+A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g:
+G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c
+t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3:
+(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2
+H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/arity/fwd.ma".
+
+theorem node_inh:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c:
+C).(\lambda (t: T).(arity g c t (ASort k n)))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0:
+nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n))))))
+(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n))))
+(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda
+(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0
+n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t:
+T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t:
+T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c:
+C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1)
+(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0
+x1) (ASort (S n0) n) H1))))) H0)))) k))).
+
+theorem arity_lift:
+ \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2
+t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1
+c2) \to (arity g c1 (lift h d t) a)))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to
+(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n:
+nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop
+h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O
+n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d))))))))
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1:
+(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall
+(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda
+(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1
+c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i
+d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def
+(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d
+(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity
+g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O
+c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1
+(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n:
+nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C
+(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i))
+u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i)
+a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h
+(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x
+d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead
+x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S
+i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0:
+T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i
+c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i))
+(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
+Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g
+a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0:
+nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g
+a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
+(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0)
+(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1
+t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0
+H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0:
+C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop
+h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d
+(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind
+nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
+(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
+H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind
+Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S
+i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0
+(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h
+(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i
+(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11)
+a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i))
+(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus
+i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h)
+(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1)
+(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
+(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
+(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1
+(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u)
+(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind
+g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4
+(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5
+b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h
+d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g
+a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1
+(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d
+u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1
+a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst)
+d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d)
+(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0))
+(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
+(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead
+a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s
+(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1
+(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s
+(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat
+Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
+A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift
+h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
+a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1
+c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d)
+t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0
+(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4))
+(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h
+d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0)
+a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1
+c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a
+H))))).
+
+theorem arity_mono:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c
+t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H:
+(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
+A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0:
+C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n)
+a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u
+a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a
+a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
+\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
+(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
+Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
+(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
+(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
+x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind
+Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _)
+\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in
+((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead
+d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr)
+u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d x0)).(let
+H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abbr)
+t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0
+t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
+(CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
+(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9)))))))
+H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
+(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
+(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
+(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g
+a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i
+c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i
+H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind
+Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind
+Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst)
+x1) H6)) in (False_ind (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c0 (CHead d (Bind Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u
+(asucc g a))).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g
+(asucc g a) a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i)
+a2)).(let H4 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2)
+(\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead
+d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
+a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
+(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
+(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
+x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind
+Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr)
+x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6))
+in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind
+Abst) x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind
+C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind
+Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst)
+x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
+c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead
+d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
+Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0
+(CHead x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def
+(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u
+H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g
+a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
+(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
+(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2
+(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall
+(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda
+(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4:
+((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3
+a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0)
+a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A
+(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u
+x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8))))
+H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda
+(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0
+u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3:
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3:
+((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3
+a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0)
+a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda
+(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda
+(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity
+g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7:
+(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u)
+t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a))
+(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8))
+a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2:
+A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g
+c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_:
+(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0
+t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity
+g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0
+H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity
+g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g
+c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x
+a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_:
+((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda
+(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2:
+A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4:
+(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u
+t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g
+a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
+a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
+(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3:
+A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2:
+(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans
+g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
+
+theorem arity_repellent:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1:
+A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c
+(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P:
+Prop).P)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1:
+A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2:
+A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g
+a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst)
+w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind
+A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4:
+A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g
+x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def
+(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y
+(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind
+Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))).
+
+theorem arity_appls_cast:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs:
+TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to
+((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl)
+vs (THead (Flat Cast) u t)) a))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads
+(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a)
+\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a)))))
+(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity
+g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u)
+(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c
+(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a:
+A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u))
+(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat
+Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1
+t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1:
+A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda
+(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat
+Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat
+Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1))
+(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g
+a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat
+Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda
+(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g
+a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))
+a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g
+a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7
+(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc
+g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g
+(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))).
+
+theorem arity_appls_abbr:
+ \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall
+(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c
+(THeads (Flat Appl) vs (TLRef i)) a)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs:
+TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads
+(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t
+(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v)
+a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop
+Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0:
+((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to
+(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda
+(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
+v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S
+i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1:
+A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity
+g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x:
+A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat
+Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat
+Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))).
+
+theorem arity_appls_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c:
+C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t:
+T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v)
+(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat
+Appl) vs (THead (Bind b) v t)) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
+(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v
+a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind
+b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v)
+t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads
+(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1
+(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead
+c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts
+(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v)
+(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in
+(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0)
+a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0
+(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda
+(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity
+g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x
+a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O)
+O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1
+(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/arity/props.ma".
+
+include "LambdaDelta-1/fsubst0/fwd.ma".
+
+include "LambdaDelta-1/csubst0/getl.ma".
+
+include "LambdaDelta-1/subst0/dec.ma".
+
+include "LambdaDelta-1/subst0/fwd.ma".
+
+include "LambdaDelta-1/getl/getl.ma".
+
+theorem arity_gen_cvoid_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
+(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to
+(\forall (P: Prop).P))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d
+(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
+(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d
+(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w
+(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P)))))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_:
+(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall
+(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall
+(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0
+(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w
+(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i)
+O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O
+w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0
+(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0
+(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d
+(Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9))))))
+(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda
+(_: ((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead
+d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v)
+\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda
+(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P:
+Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq
+nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat
+i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let
+H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0
+(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead
+d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Void)
+u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7))
+in (False_ind P H9)))))) (subst0_gen_lref w v i0 i H4))))))))))))))))))
+(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2:
+((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d
+(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to
+(\forall (P: Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d:
+C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d
+(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
+(\forall (P: Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0)
+v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind
+b) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq
+T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0
+t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T
+(\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i
+w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v
+(THead (Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9
+P)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u
+t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda
+(t2: T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b)
+i) w t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u
+x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i)
+(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d
+(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u
+x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9
+P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6)))))))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl
+i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w:
+T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
+T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P:
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
+(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2:
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind
+Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
+(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
+(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2)))
+(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s
+(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind
+Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8
+P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda
+(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda
+(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1
+d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i
+H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
+(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0
+t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall
+(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall
+(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind
+Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead
+(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T
+(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w
+u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0
+t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2
+t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T
+v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda
+(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0
+i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2:
+T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl)
+i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2)))
+(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s
+(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
+(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w
+u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
+P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5)))))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u
+(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
+(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
+T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P:
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
+(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2:
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat
+Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
+(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
+(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2)))
+(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s
+(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
+(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w
+u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
+P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5))))))))))))))))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
+t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2:
+A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d
+u i H3 w v H4 P)))))))))))))))) c t a H))))).
+
+theorem arity_gen_cvoid:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
+(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in
+(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i
+v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i
+v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T
+t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift
+(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3:
+(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u
+(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))
+(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda
+(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i
+x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v)))))
+(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x
+(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
+
+theorem arity_fsubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
+c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u
+c1 t1 c2 t2) \to (arity g c2 t2 a))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
+d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2
+t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n:
+nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i
+c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1:
+(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort
+n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u
+(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i
+u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3:
+(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i
+u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c
+c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0:
+C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2
+(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0
+i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2
+(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c
+c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n)))
+(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n)
+t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c
+c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n)
+t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g
+c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
+Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2:
+((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2
+t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef
+i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in
+(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))
+(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i)
+t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2)
+(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i)
+t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0
+(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq
+nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat
+i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O
+u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind
+Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in
+((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d
+(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
+i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d d1)).(let H16
+\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H12
+u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let
+H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u)))
+H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i
+H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7)))
+H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind
+(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq
+T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i)
+(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0)
+(\lambda (H9: (lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8
+(CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abbr) u))
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w:
+T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i))
+u0 e1 e2))))))) (arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d
+(Bind Abbr) u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n:
+nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)))
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
+in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda
+(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind
+x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def
+(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u)
+(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3
+(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0
+(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in
+(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def
+(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18)
+in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind
+x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i
+c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0
+(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead
+d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr)
+(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda
+(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b)
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq
+C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2
+(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1
+x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u)
+(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
+b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let
+H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t)))
+H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus
+i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u
+i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind
+Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11))
+(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
+x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i)
+(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)))
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
+in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0]))
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H19
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr
+x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
+x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
+H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
+H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr)
+(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u
+(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0
+d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
+(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
+d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
+(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
+(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
+T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
+nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind
+Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in
+((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d
+(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
+i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17
+\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13
+u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2))
+H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0))
+(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr)
+u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u
+i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0
+H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6))
+H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1:
+C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0))
+\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g
+c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0:
+nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2:
+C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x
+\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in
+(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i)
+t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c
+c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef
+i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0)
+(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind
+C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2
+(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
+T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n
+c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d
+(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in
+(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d
+(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c
+(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2
+H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c
+c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0)
+(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c
+c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0
+(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def
+(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in (or4_ind
+(getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0)
+(\lambda (H11: (getl i c2 (CHead d (Bind Abst) u))).(let H12 \def (eq_ind nat
+(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1
+(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d
+(Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i)))
+(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11:
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda
+(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2
+(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2
+x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
+b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let
+H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3))
+H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead
+c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3
+i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind
+Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11))
+(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b)
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq
+C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2
+(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1
+x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
+b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u)
+(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let
+H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t)))
+H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus
+i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u
+i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind
+Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11))
+(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
+x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i)
+(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0)))
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
+in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0]))
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H19
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst
+x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
+x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
+H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
+H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst)
+(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u
+(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0
+d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
+(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
+d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
+(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
+(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
+T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
+nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind
+Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr)
+u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12))
+in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10)))
+(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall
+(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to
+(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall
+(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b)
+u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead
+(Bind b) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u
+t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0
+(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0
+c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2))
+(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind
+b) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2)
+(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0
+(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2))
+(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda
+(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b)
+u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
+u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0
+u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2))
+(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t
+a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b
+c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S
+i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u
+x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s
+(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity
+g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c
+(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda
+(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t
+x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2))
+(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13))
+x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind
+b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both
+(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0)
+(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head
+(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead
+(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t)
+t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind
+b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u
+t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0
+i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c
+(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u))))
+t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2)
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2)
+(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead
+(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
+u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2
+T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0
+u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2))
+(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13
+c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u
+(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t
+(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x)
+(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda
+(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2
+(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))
+(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u
+x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind
+b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2
+d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c
+(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10
+u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b)
+i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)))
+(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2
+(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14:
+(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda
+(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0
+(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind
+b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5)
+(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1
+H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2
+H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8))
+H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1:
+C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0))
+\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g
+c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1:
+C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead
+d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0
+(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda
+(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1
+(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i
+u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2
+(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq
+C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind
+Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u
+t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land
+(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2)
+(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2))
+(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t)
+t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2:
+T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u
+t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2
+(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind
+Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
+T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))
+(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead
+(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind
+Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x
+a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i)
+u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i
+u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq
+T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0
+t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1
+a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u
+x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead
+(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g
+c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u)
+c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind
+Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2
+H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity
+g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T
+t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda
+(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0
+x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1
+d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0)
+(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10))
+(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7:
+(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T
+(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2))
+(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0
+c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0
+(AHead a1 a2))) (arity_head g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c
+u c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u)
+c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind
+Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind
+Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda
+(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c
+c2))).(land_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2)
+(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u
+t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2:
+T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3:
+T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 (AHead a1 a2)) (\lambda
+(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12:
+(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0:
+T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind
+Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr)
+u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind
+Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2
+H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1
+a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u
+x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead
+(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head
+g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S
+i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u)
+(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2
+(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2
+(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1
+u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0)
+(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10))
+(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6)))))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u
+a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t:
+T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3:
+((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
+t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat
+Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2)
+(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t)
+t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2)
+(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2)
+(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0
+(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c
+c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c
+(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T
+t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T
+(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0
+(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
+t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2
+(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
+(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead
+(Flat Appl) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat
+Appl) x t) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0
+i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda
+(H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda
+(t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq
+T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0
+t t3)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat
+Appl) u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T
+(THead (Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u
+a1 H0 x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10))
+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity
+g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
+(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
+(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0
+(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c
+t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9))
+c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0
+i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)
+(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda
+(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst
+i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
+H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2)
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2)
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+(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10:
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+Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda
+(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i
+u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T
+t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
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+(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x)
+(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2
+u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c
+t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
+Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1
+(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head
+(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g
+a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
+a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3:
+((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
+t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat
+Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2)
+(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t)
+t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2)
+(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0
+(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c
+c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c
+(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T
+t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T
+(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0
+(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
+t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2
+(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
+(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead
+(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat
+Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0
+i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10:
+(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2
+(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
+t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat
+Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T
+(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u
+a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10))
+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity
+g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
+(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
+(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0
+(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t
+x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2
+H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i
+u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)
+(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda
+(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst
+i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
+H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead
+(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10:
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2:
+T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat
+Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda
+(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0
+c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda
+(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x)
+(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2
+u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t
+x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
+Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i
+u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t
+t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t:
+T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall
+(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr)
+u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity
+g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda
+(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1
+(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u
+c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def
+H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2)
+(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2
+t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c
+c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda
+(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2))
+(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2)
+c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind
+(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t
+t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g
+c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2
+H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u
+c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2)
+(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c
+c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7
+c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))).
+
+theorem arity_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c
+t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
+d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2
+a)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H:
+(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
+(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u
+c t1 t2 H1)))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/A/defs.ma".
+
+include "LambdaDelta-1/G/defs.ma".
+
+definition asucc:
+ G \to (A \to A)
+\def
+ let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n)
+\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h)
+\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g
+a2))]) in asucc.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/asucc/defs.ma".
+
+theorem asucc_gen_sort:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A
+(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
+nat).(eq A a (ASort h0 n0)))))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind
+(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda
+(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0
+n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0
+with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0
+n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A
+(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1)))))))
+(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat
+nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0
+n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1))
+\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0
+n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2
+\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda
+(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2)))))))
+a)))).
+
+theorem asucc_gen_head:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A
+(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind
+(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3:
+A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3))))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc
+g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g
+(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead
+a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0))
+H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0)))
+(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda
+(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0:
+A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
+a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1)
+n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0:
+A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
+a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2)
+(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda
+(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A
+(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1
+a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead
+a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
+(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in
+((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4]))
+(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let
+H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4))
+\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A
+a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A
+(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A
+a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead
+a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5)))
+(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7
+\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to
+(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4
+(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda
+(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda
+(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead
+a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3
+(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4))))
+H2))))))) a)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/defs.ma".
+
+definition cimp:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall
+(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C
+(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/cimp/defs.ma".
+
+include "LambdaDelta-1/getl/getl.ma".
+
+theorem cimp_flat_sx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v)
+c)))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f)
+v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat
+f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2
+(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind
+b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1
+(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c
+(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b)
+w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v)
+(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind
+b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind
+b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w)))
+d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))).
+
+theorem cimp_flat_dx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f)
+v))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind
+b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2
+(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
+
+theorem cimp_bind:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
+(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
+(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w:
+T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1
+(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v)
+(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b)
+v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v)
+(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
+\Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind
+b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1
+(Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in
+C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead
+c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
+(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead
+c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
+(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0
+b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda
+(d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B
+b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v)
+(CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2
+(Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4))))
+H3)) H2))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c1 (Bind b) v)
+(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl h0 (CHead c2 (Bind
+b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind
+b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0)
+(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x
+in (ex_ind C (\lambda (d2: C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C
+(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))))
+(\lambda (x: C).(\lambda (H3: (getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C
+(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))
+x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w) H3 v)))) H2)))))) h
+H0)))))))))).
+
+theorem cimp_getl_conf:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
+(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w))
+\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind b) w)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
+(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def
+H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C
+(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall
+(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4:
+C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x
+(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3:
+C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0))
+\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0))))))))))
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0:
+B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h
+d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1
+(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0
+(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in
+(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2
+(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind
+b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0
+(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind
+b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat
+(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n:
+nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0
+(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def
+(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind
+b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro
+C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind
+b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+inductive clear: C \to (C \to Prop) \def
+| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e
+(Bind b) u) (CHead e (Bind b) u))))
+| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f:
+F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/clear/fwd.ma".
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+theorem drop_clear:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to
+(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead
+e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
+c2))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i:
+nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda
+(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind
+(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v)))))
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda
+(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat
+O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_:
+B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort
+n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall
+(i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda
+(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k
+t) c2)).(K_ind (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead
+e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
+c2))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) i) O c
+c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v:
+T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_:
+B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c
+t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2
+\def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda
+(v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e:
+C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda
+(e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v)))))
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda
+(x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1
+(Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda
+(b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e
+(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
+c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2))))
+k (drop_gen_drop k c c2 t i H0))))))))) c1).
+
+theorem drop_clear_O:
+ \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c
+(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1
+e2) \to (drop (S i) O c e2))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1:
+C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
+C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2))))))))
+(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort
+n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_:
+(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O
+(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1:
+C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
+C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0
+e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u:
+T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0:
+K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0
+k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t)
+(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in
+C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _)
+\Rightarrow c1])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t)
+(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_:
+K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1
+(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b)
+u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow
+t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0
+(CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq
+C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: C).(drop i O c1 e2)) H1 c0
+H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2))
+(drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f:
+F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b)
+u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind
+b) u) t H2) e2 i H1) t))) k H0))))))))))) c)).
+
+theorem drop_clear_S:
+ \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop
+h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear
+x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1
+(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))
+\def
+ \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
+C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda
+(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b)
+u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1:
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
+C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k
+t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear
+(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1
+(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C
+(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1:
+C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift
+h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k
+(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead
+c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind
+(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h
+(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d)
+t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))
+(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind
+b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u)
+(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in
+((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in
+K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
+\Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t)
+(clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C
+T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead
+c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda
+(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0:
+T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d)
+t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2))))
+(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind
+b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda
+(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda
+(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind
+b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda
+(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind
+b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x
+(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f:
+F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda
+(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u
+(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1:
+C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d)
+t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))
+(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d
+u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear
+(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d
+u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b)
+(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1
+H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/clear/defs.ma".
+
+theorem clear_gen_sort:
+ \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda
+(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_:
+C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c:
+C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let
+H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
+\Rightarrow True])) I (CSort n) H1) in (False_ind P H2)))))) (\lambda (e:
+C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (_: (((eq C e (CSort
+n)) \to P))).(\lambda (f: F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e
+(Flat f) u) (CSort n))).(let H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
+(False_ind P H4))))))))) y x H0))) H)))).
+
+theorem clear_gen_bind:
+ \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
+(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
+(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda
+(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0:
+(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e
+(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b)
+u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow
+c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H3 \def
+(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def
+(f_equal C T (\lambda (e1: C).(match e1 in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind
+b0) u0) (CHead e (Bind b) u) H1) in (\lambda (H5: (eq B b0 b)).(\lambda (H6:
+(eq C e0 e)).(eq_ind_r T u (\lambda (t: T).(eq C (CHead e0 (Bind b0) t)
+(CHead e0 (Bind b0) t))) (eq_ind_r C e (\lambda (c: C).(eq C (CHead c (Bind
+b0) u) (CHead c (Bind b0) u))) (eq_ind_r B b (\lambda (b1: B).(eq C (CHead e
+(Bind b1) u) (CHead e (Bind b1) u))) (refl_equal C (CHead e (Bind b) u)) b0
+H5) e0 H6) u0 H4)))) H3)) H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda
+(_: (clear e0 c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c
+e0)))).(\lambda (f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat
+f) u0) (CHead e (Bind b) u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (CHead e (Bind b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f)
+u0)) H4))))))))) y x H0))) H))))).
+
+theorem clear_gen_flat:
+ \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
+(CHead e (Flat f) u) x) \to (clear e x)))))
+\def
+ \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
+(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda
+(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda
+(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead
+e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f)
+u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
+_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1)
+in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda (e0:
+C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0
+(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0:
+T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4
+\def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0])) (CHead e0
+(Flat f0) u0) (CHead e (Flat f) u) H3) in ((let H5 \def (f_equal C F (\lambda
+(e1: C).(match e1 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow
+f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) with
+[(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0)
+u0) (CHead e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1:
+C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u)
+H3) in (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let H9 \def
+(eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to (clear e
+c))) H2 e H8) in (let H10 \def (eq_ind C e0 (\lambda (c0: C).(clear c0 c)) H1
+e H8) in H10))))) H5)) H4))))))))) y x H0))) H))))).
+
+theorem clear_gen_flat_r:
+ \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x
+(CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H:
+(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e
+(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y:
+C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0:
+C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat
+f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat
+f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda
+(H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to
+P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f)
+u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f)
+u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda
+(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C
+(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
+
+theorem clear_gen_all:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind
+(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0:
+B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0
+(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e:
+C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b)
+u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b)
+u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c
+(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c
+(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C
+(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B
+C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind
+x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0)
+x2))) c H3)))))) H2)))))))) c1 c2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/clear/fwd.ma".
+
+theorem clear_clear:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
+(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear
+(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c:
+C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2
+c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear
+(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to
+(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t)
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0))
+(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f:
+F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c
+c2 t H1)))) k H0))))))) c1).
+
+theorem clear_mono:
+ \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c
+c2) \to (eq C c1 c2)))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n:
+nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2:
+C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1
+c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t)
+c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2)
+\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b)
+t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0
+(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t)
+(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0
+(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t
+H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t)
+c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f
+c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
+
+theorem clear_trans:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c
+c2) \to (clear c1 c2)))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2:
+C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n)
+c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t)
+c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0:
+K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b:
+B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0
+(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c
+c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead
+c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3)))))
+(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c
+c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
+
+theorem clear_ctail:
+ \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1
+(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k
+u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
+K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b)
+u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H:
+(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1:
+T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1
+c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n
+H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind
+b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear
+(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2))))
+k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2:
+T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1:
+T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear
+(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1:
+T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to
+(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2))))
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind
+b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t)
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_:
+K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2
+(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b)
+u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0)
+\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t)
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b
+b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead
+(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r
+C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail
+k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0
+u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail
+k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1:
+(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1
+c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead
+c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)).
+
+theorem clear_cle:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
+(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda
+(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O)))))
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight
+c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
+C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear
+(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t)))))
+(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C
+(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c)
+(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c
+c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t)
+c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2
+(clear_gen_flat f c c2 t H1))))) k H0))))))) c1).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+include "LambdaDelta-1/s/defs.ma".
+
+definition clen:
+ C \to nat
+\def
+ let rec clen (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O |
+(CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/clen/defs.ma".
+
+include "LambdaDelta-1/getl/props.ma".
+
+theorem getl_ctail_clen:
+ \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n:
+nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
+\def
+ \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex
+nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n)
+(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O
+(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b
+(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl
+(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k:
+K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl
+(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat
+(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0)
+(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen
+c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0:
+K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t
+c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat
+(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0)
+t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail
+(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f:
+F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t
+c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b)
+t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))).
+
+theorem getl_gen_tail:
+ \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall
+(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2
+(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
+\def
+ \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i
+(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
+c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4
+nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
+n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0:
+nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n)
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort
+n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2))
+(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead
+(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear
+(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e:
+C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_:
+nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
+nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead
+(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead
+(CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2)
+u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow
+(match k0 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
+(Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0)
+u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2
+(Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n)
+(CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5:
+(eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C
+(\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O
+(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O))
+(\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2))
+(\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or
+(ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e:
+C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq
+nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq
+T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0
+(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0)
+u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat
+(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1)))
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort
+n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0)
+u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat
+(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0)))
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort
+n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K
+(Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq
+C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0))
+(refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2))
+H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead (CSort n) (Flat f) u1)
+(CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n
+(clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda
+(e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n)
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda
+(_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) k u1)
+(CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 (CHead
+(CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
+c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b)
+u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: nat).(eq K k
+(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort
+n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind
+b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k
+(CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda (e: C).(eq C
+c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b)
+u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K
+k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2
+(CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: ((\forall (i:
+nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda
+(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b)
+u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq
+K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2
+(CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(nat_ind
+(\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b)
+u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
+C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))
+(\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b)
+u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2
+(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat
+(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
+n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind
+b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
+(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c)
+(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1))
+in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
+(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow (match
+k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
+\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t)
+(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2
+(Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k
+u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda
+(H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c
+(Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O
+(s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b
+(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4
+nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_:
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
+C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i0:
+nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda
+(e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i0 c (CHead e (Bind b)
+u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 (clen c))) (\lambda (_: nat).(eq
+K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0
+(CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda
+(c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e:
+C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda
+(_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort
+n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1
+e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2))))
+(ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_:
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
+C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1
+c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e
+(Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6))
+b0 H5) t H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead
+(CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O
+(getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl
+(CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t
+H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda
+(e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat
+O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
+u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C
+c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2
+(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind
+C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead
+e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4
+nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
+K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2
+(CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1
+x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1
+x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e)))
+(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4
+nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
+K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0
+(CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail
+k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b)
+u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda
+(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n:
+nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C
+(CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t)
+(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x
+(Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_:
+nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat
+(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or
+(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O
+(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
+nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
+(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0:
+nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind
+b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort
+x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq
+C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C
+(\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O
+(CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq
+nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
+(_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))))))
+(eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort
+x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
+(Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
+(\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1))
+(\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda
+(e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O
+(CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq
+nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort
+n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
+(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1))
+(\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b))
+(refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3))
+H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2)
+H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead (CTail k u1 c) k0 t)
+(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
+e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
+(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort
+n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2
+(Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead
+c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (e:
+C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind
+b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_:
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
+nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
+e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4
+nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K
+k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2
+(CSort n0))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c
+(CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
+(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k
+(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort
+n0))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5:
+(getl (r k0 n) c (CHead x (Bind b) u2))).(let H6 \def (eq_ind C c2 (\lambda
+(c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0
+(CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) (CTail k u1 x) H4) in (let H7
+\def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t)
+(CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1
+e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
+(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort
+n0))))))) H0 (CTail k u1 x) H4) in (eq_ind_r C (CTail k u1 x) (\lambda (c0:
+C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl
+(S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
+nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl
+(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x)
+(CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1
+e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x
+(refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H5 t)))
+c2 H4)))))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n)
+(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
+u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))).(ex4_ind nat (\lambda (_:
+nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or
+(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n)
+(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S
+n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0:
+nat).(\lambda (H4: (eq nat (r k0 n) (clen c))).(\lambda (H5: (eq K k (Bind
+b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(let H8
+\def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0
+(Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1)
+(CSort x0) H7) in (let H9 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead
+(CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e:
+C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c))))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n0: nat).(eq C c0 (CSort n0))))))) H0 (CSort x0) H7) in (eq_ind_r C (CSort
+x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e)))
+(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
+(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k
+(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort
+n0)))))) (let H10 \def (eq_ind_r T u2 (\lambda (t0: T).((getl n (CHead (CTail
+k u1 c) k0 t) (CHead (CSort x0) (Bind b) t0)) \to (or (ex2 C (\lambda (e:
+C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t)
+(CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen
+c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0))
+(\lambda (n0: nat).(eq C (CSort x0) (CSort n0))))))) H9 u1 H6) in (let H11
+\def (eq_ind_r T u2 (\lambda (t0: T).(getl (r k0 n) (CTail k u1 c) (CHead
+(CSort x0) (Bind b) t0))) H8 u1 H6) in (eq_ind T u1 (\lambda (t0: T).(or (ex2
+C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S
+n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat
+(S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let
+H12 \def (eq_ind K k (\lambda (k1: K).((getl n (CHead (CTail k1 u1 c) k0 t)
+(CHead (CSort x0) (Bind b) u1)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort
+x0) (CTail k1 u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind
+b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_:
+nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0:
+nat).(eq C (CSort x0) (CSort n0))))))) H10 (Bind b) H5) in (let H13 \def
+(eq_ind K k (\lambda (k1: K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0)
+(Bind b) u1))) H11 (Bind b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or
+(ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b)))
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
+n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e:
+C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n)
+(CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S
+n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_:
+nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1))))))
+(eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C
+(CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t)
+(CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0))
+(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1))
+(\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C
+(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
+n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_:
+nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0:
+nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K
+(Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r
+k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) H2)))))) i))))))
+c1)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+inductive cnt: T \to Prop \def
+| cnt_sort: \forall (n: nat).(cnt (TSort n))
+| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt
+(THead k v t))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/cnt/defs.ma".
+
+include "LambdaDelta-1/lift/fwd.ma".
+
+theorem cnt_lift:
+ \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i
+d t)))))
+\def
+ \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i:
+nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i:
+nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0))
+(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0:
+T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d:
+nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0))
+(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k
+(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/getl.ma".
+
+include "LambdaDelta-1/csuba/props.ma".
+
+include "LambdaDelta-1/arity/props.ma".
+
+include "LambdaDelta-1/csubv/getl.ma".
+
+theorem csuba_arity:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c
+c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall
+(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda
+(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in
+(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda
+(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d
+x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc
+g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u
+(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def
+(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc
+g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6:
+(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst
+g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0
+(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u
+(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i)
+x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u
+(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
+(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda
+(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4:
+((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0
+a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2
+u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind
+b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2:
+C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2)
+\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
+(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1
+a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1
+(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g
+a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2:
+C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2
+H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
+(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity
+g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
+C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
+H2)))))))))) c1 t a H))))).
+
+theorem csuba_arity_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2
+t a))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0
+a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_:
+(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u
+a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to
+(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2
+c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i
+H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
+u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity
+g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))
+(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x
+(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf
+c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda
+(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2
+(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let
+H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0
+(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1
+(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in
+C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono
+c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead
+d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0
+(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda
+(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c
+(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda
+(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def
+(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def
+(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16)
+in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13))))))))
+H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
+u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_:
+(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10:
+(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7
+x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void)
+x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c
+H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2:
+C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i
+c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2:
+C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c
+(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c
+(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d
+(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2
+(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g
+a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to
+(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2
+c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i
+H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0)
+(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2
+(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst)
+u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4
+Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda
+(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2
+(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let
+H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0
+(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1
+(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in
+C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono
+c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead
+d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0
+(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda
+(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c
+(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda
+(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def
+(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def
+(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16)
+in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13))))))))
+H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl
+i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let
+H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in
+(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2:
+C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef
+i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda
+(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d
+(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3)
+(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in
+(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d
+(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2
+(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
+(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0
+a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to
+((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2:
+C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b
+H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c
+H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g
+a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to
+(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
+(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2:
+C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst)
+u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2
+c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3
+(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u)
+(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda
+(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0
+(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda
+(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4
+H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda
+(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2
+c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0:
+T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g
+c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda
+(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2
+H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
+C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda
+(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g
+c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2
+H2))))))))))) c1 t a H))))).
+
+theorem arity_appls_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c
+v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t:
+T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs
+(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead
+(Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H:
+(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g
+a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr)
+v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead
+(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead
+(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B
+Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in
+(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead
+c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u
+t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g
+(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t)
+a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t
+a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H)
+(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst)
+Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1:
+((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
+a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
+Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def
+(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2)
+in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
+(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0
+x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
+(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3)))))))
+vs))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/defs.ma".
+
+include "LambdaDelta-1/clear/fwd.ma".
+
+theorem csuba_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2))
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2))
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
+(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u)
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear
+(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind
+b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2))))
+(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda
+(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g
+c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))))
+(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def
+(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g
+e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1
+e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x:
+C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C
+(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f)
+u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
+C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall
+(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda
+(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3:
+(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1)
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2:
+C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba
+g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b)
+u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2)
+(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3))))))))))))
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_:
+((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2))
+(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda
+(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u
+a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t)
+e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u)
+e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2))
+(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr)
+u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1
+(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
+
+theorem csuba_clear_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1))
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2
+c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear
+c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1))
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
+(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u)
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear
+(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind
+b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2))))
+(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda
+(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g
+c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3))))
+(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def
+(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g
+e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2
+e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x:
+C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C
+(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f)
+u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
+C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall
+(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda
+(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3:
+(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2)
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2:
+C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2:
+C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3
+(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1
+u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3))))))))))))
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_:
+((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1))
+(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda
+(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u
+a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u)
+e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t)
+e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u)))
+(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst)
+t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1
+(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/arity/defs.ma".
+
+inductive csuba (g: G): C \to (C \to Prop) \def
+| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
+| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
+(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
+| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
+(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u:
+T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind
+Abbr) u))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/fwd.ma".
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+theorem csuba_drop_abbr:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
+O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
+c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))))
+(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1
+(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0:
+(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in
+(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba
+g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2
+(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x
+(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda
+(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3))))
+H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1:
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S
+n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n)
+O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind
+Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3:
+(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n)
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr)
+u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u:
+T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall
+(c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead
+d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n)
+O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba
+g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c
+(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr)
+u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop
+(r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6:
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def
+(csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2:
+C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind
+Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t)
+(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x
+H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
+d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr)
+u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind
+Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
+(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead
+x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) (\lambda (H5:
+(csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O
+c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g c c2 t H5) in
+(let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2
+(Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2:
+C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind
+Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t)
+(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x
+H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
+d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr)
+u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind
+Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
+(CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead
+x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O
+c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0
+(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t
+(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind
+Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u
+H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x
+(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat
+(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0
+O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0
+(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7)))))
+(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r
+(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g
+c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C
+c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead
+x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def
+(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead
+x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal
+nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0:
+nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n
+x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3
+H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t)
+c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let
+H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n)
+O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f)
+x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1)
+(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0
+H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
+d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr)
+u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n)
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
+d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10))))
+H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n
+H1)))))))))))) c1)))) i).
+
+theorem csuba_drop_abst:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i
+O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n
+O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1:
+T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g:
+G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1
+(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl
+c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1
+H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3:
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
+O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x
+(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind
+Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst)
+u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind
+Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2
+H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind
+Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1
+(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind
+Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr)
+x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind
+Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6
+H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H:
+((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1
+(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to
+(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop
+(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1:
+C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind
+Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0)
+c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O)
+(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda
+(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S
+n) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5)))))
+(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead
+d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to
+(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda
+(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst)
+u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t)
+c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n)
+O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c
+(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst)
+u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop
+(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g
+(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead
+d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7
+\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t)))
+(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr)
+t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t)
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
+(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead
+x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def
+(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda
+(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14)
+in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst)
+u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12:
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1
+x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0
+x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12
+(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr)
+n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2
+H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda
+(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def
+(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g
+c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity
+g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst)
+t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x:
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c
+x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g
+x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
+Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14:
+(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst)
+u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst)
+n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13:
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1
+x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0
+x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13
+(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
+(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst)
+n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2
+H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind
+Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g
+x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1)
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda
+(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
+x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
+(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead
+x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal
+nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0:
+nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in
+(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x
+(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15:
+(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1
+x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3
+x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4)))
+H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n)
+O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5
+(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17
+H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g
+(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead
+d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7
+\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2:
+T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2:
+C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
+O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c
+x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g
+x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1
+(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda
+(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
+(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x:
+C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13:
+(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst)
+u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n
+x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12:
+(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1
+x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3
+x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4)))
+H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n)
+O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5
+(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14
+H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f:
+F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r
+(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c
+c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda
+(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0
+(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f)
+x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2:
+C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda
+(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10:
+(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1
+x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1)
+H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10:
+(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1
+x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2
+x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f)
+x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind
+Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2
+(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
+
+theorem csuba_drop_abst_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
+O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
+c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n
+O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1:
+C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind
+Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2
+c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1
+(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x
+\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or
+(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst)
+u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u)
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O
+(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x
+(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind
+Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x
+(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void)
+x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1)
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O
+(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0
+(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind
+Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H:
+((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1
+(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or
+(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop
+(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n)
+O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind
+Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
+(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq
+nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n)
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
+O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u)
+H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u:
+T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall
+(c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop
+(S n) O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda
+(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0:
+K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst)
+u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda (H3:
+(csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c
+(CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c
+(Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to
+(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr)
+t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst)
+u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in
+(or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda
+(d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
+(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t)))
+(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
+O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr)
+t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t)
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x
+H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind
+Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
+g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
+(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14:
+(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind
+Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
+g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr)
+t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
+(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12:
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda
+(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void)
+x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1
+(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12))
+H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc
+g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))
+(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g
+x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t
+x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C
+(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
+O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda
+(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16:
+(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind
+Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
+x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14))
+(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void)
+x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4
+(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14))
+H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9:
+(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r
+C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2:
+C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let
+H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O
+x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C
+(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
+(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x
+d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
+x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop
+(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda
+(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
+x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12))
+H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst)
+t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst)
+u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in
+(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda
+(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba
+g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2
+(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x
+(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H
+c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
+O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
+Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst)
+u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
+O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S
+n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u)
+H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
+Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead
+x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1
+(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12))
+H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9:
+(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r
+C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2:
+C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let
+H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O
+x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C
+(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
+(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x
+d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
+x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop
+(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda
+(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
+x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12))
+H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void)
+t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst)
+u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2:
+C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq
+C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C
+(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S
+n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H
+c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
+O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
+Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst)
+u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
+O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S
+n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u)
+H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
+Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead
+x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1
+(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11))
+H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2
+(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind
+Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def
+H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
+O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0
+(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f)
+x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0
+H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
+(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat
+f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst)
+u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S
+n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10
+x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0
+(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
+(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3
+(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9))
+H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n
+H1)))))))))))) c1)))) i).
+
+theorem csuba_drop_abbr_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i
+O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba
+g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
+(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c:
+C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1
+(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in
+(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
+(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x
+d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C
+(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2:
+C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr)
+u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl
+(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g
+x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1
+u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C
+(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2:
+C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst)
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2
+(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda
+(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C
+c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C
+(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop
+O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))
+(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind
+Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T
+(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0
+x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2)))))))))))
+(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall
+(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall
+(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0:
+nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0)
+(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_:
+(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort
+n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1)
+(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let
+H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
+in (False_ind (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5)))))
+(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead
+d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop
+(S n) O (CHead c k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda
+(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0:
+K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr)
+u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda
+(b: B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop
+(r (Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0:
+B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead
+d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6:
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
+(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or3_ind (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
+g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g c t a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+c)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda
+(d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2
+(Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq
+C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C
+(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop
+(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S
+n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr)
+t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14:
+(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
+(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda
+(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0
+(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0
+x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n
+x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12:
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void)
+x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind
+Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9))))
+H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind
+Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc
+g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1)
+(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda
+(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
+C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst)
+x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
+(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x
+d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst)
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0
+(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
+A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16:
+(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18:
+(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
+x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5
+(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17
+H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop
+n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void)
+x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind
+Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2
+H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void)
+x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1)
+(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda
+(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
+C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void)
+x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
+(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x
+d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0
+(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14:
+(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16:
+(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
+x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15
+H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop
+n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
+x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind
+Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9)))))
+H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda
+(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
+(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba
+g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8:
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2:
+C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq
+C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C
+(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop
+(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S
+n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst)
+t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14:
+(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
+(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda
+(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0
+(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0
+x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n
+x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12:
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void)
+x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
+(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind
+Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9))))
+H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void)
+x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1)
+(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda
+(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
+C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void)
+x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
+(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x
+d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0
+(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14:
+(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16:
+(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
+x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15
+H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop
+n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
+x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind
+Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9)))))
+H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda
+(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
+(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c))
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x:
+C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x
+c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x
+H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0
+(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda
+(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
+x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11))
+(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0
+(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0
+x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n
+x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11:
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void)
+x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind
+Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8))))
+H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat
+f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr)
+u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in
+(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda
+(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0
+c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0
+H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda
+(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr)
+u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop
+(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9:
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3:
+T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst)
+x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g
+x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4
+(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12
+H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
+(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2
+(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda
+(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
+x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0
+(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k
+H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/defs.ma".
+
+theorem csuba_gen_abbr:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
+(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
+(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u)
+(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda
+(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0)
+(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3)
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C
+c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0
+c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr)
+u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2
+(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1
+(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
+\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr)
+u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2)
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5)))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_:
+(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t:
+T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0:
+T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst)
+t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr)
+u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0)
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6))))))))))))
+y c H0))) H))))).
+
+theorem csuba_gen_void:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
+(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void)
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda
+(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2)))))
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))
+(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0:
+C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind
+Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C
+T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2
+(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u)
+(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3)
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
+T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda
+(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind
+Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda
+(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind
+C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g
+c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))
+Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6))))
+H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1
+(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in
+((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead
+c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1
+d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind
+Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3:
+T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C
+c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda
+(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead
+d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba
+g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9)))))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc
+g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1
+(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2)))))
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))
+H6)))))))))))) y c H0))) H))))).
+
+theorem csuba_gen_abst:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
+(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst)
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
+C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y
+c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind
+Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C
+(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda
+(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2:
+C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
+(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1
+(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
+\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
+(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
+(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2:
+C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
+(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst)
+(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1
+(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2
+d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1
+H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst)
+u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2
+(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))
+c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b
+Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind
+Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind
+Void) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
+Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind
+Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind
+b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3
+C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind
+b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1
+(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc
+g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t)
+(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t |
+(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind
+Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t
+(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def
+(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let
+H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1))
+\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8)
+in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in
+(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
+g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
+A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g
+a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
+a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4))))))))
+H6)))))))))))) y c H0))) H))))).
+
+theorem csuba_gen_flat:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
+(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0:
+(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u)
+(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C
+c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C
+(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1
+(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1
+(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3:
+T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda
+(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat
+f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))
+H6)))))))))))) y c H0))) H)))))).
+
+theorem csuba_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
+(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
+(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b:
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1)
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1
+(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1
+H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9)
+in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind
+b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b)
+u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2))
+H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3
+B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
+e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t
+(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5:
+(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind
+Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda
+(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind
+Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t
+| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind
+b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1
+e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a)))
+H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc
+g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let
+H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15
+\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
+e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7))
+H6)))))))))))) y c2 H0))) H)))))).
+
+theorem csuba_gen_abst_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
+(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
+(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u)
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
+C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y:
+C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1:
+C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C
+c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n:
+nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def
+(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda
+(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
+H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C
+(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k
+u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
+(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3)
+in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1
+(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C
+c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C
+(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))
+(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C
+(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let
+H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to
+(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g
+c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind
+Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind
+Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda
+(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1
+(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not
+(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead
+c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2
+| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind
+Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1
+(Bind Abst) u) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
+\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in
+(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def
+(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11
+\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to
+(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1
+(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g
+c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind
+Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1)
+(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1
+(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1))
+H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
+(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or
+(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc
+g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C
+(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead
+d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead
+c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind
+Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
+g d2 d1))))) H6)))))))))))) c y H0))) H))))).
+
+theorem csuba_gen_void_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
+(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
+Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
+(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u)
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda
+(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0)
+(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3)
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1
+c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void)
+u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1
+(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1
+(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in
+((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in
+((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead
+c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b
+Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0:
+B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda
+(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9)
+in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in
+(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_:
+False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13)))))))
+H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C
+(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead
+d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1
+(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))
+H6)))))))))))) c y H0))) H))))).
+
+theorem csuba_gen_abbr_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr)
+u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda
+(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda
+(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C
+(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind
+Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2
+C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C
+c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
+(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
+T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t)
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2:
+C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C
+(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba
+g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1
+(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
+(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
+(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b:
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr)
+u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b)
+u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda
+(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
+Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void)
+u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in
+(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in
+(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0)
+(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a:
+A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq
+C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2:
+C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void)
+u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C
+(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t:
+T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u:
+T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr)
+u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
+(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind
+Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0)
+\Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in
+(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity
+g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g
+c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g
+a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1
+a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8)
+in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in
+(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
+(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1
+u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10))))))))
+H6)))))))))))) c y H0))) H))))).
+
+theorem csuba_gen_flat_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
+(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0:
+(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
+(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C
+c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C
+(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1
+(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1
+(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3:
+T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda
+(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat
+f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
+(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+H6)))))))))))) c y H0))) H)))))).
+
+theorem csuba_gen_bind_rev:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
+(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
+(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2
+y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u)
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b:
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let
+H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3
+(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B
+(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3
+(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) u2) (CHead
+e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3
+e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1
+H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C
+c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1))
+H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3
+B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
+e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t
+(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5:
+(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind
+Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda
+(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind
+Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u
+| (CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind
+b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3
+e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8)
+in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10)
+in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1)
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
+C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
+(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3
+(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1
+(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2)
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
+e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2)
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))
+Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7))
+H6)))))))))))) c2 y H0))) H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/drop.ma".
+
+include "LambdaDelta-1/csuba/clear.ma".
+
+include "LambdaDelta-1/getl/clear.ma".
+
+theorem csuba_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g
+c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u)))
+(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x:
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
+Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1
+(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda
+(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1
+(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3:
+(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1
+(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to
+((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2:
+C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda
+(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0
+(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
+(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
+Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u)
+t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda
+(c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda
+(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
+(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr
+H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c
+(Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15
+g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1:
+C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18:
+(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1
+(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18))))
+H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead
+x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind
+Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c
+(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n
+O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead
+x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10
+\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t)
+(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0
+(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6)
+f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead
+d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1
+(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))
+(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u)
+x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u
+H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))
+(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda
+(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2
+c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl
+O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3
+(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16)))))
+H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1:
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2)
+\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O
+x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1
+c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B
+C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind
+b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead
+x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3:
+C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2)
+x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def
+(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C
+(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2:
+C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15:
+(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x
+\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B
+C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
+x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda
+(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19:
+(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16
+(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind
+C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22:
+(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1
+x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead
+x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11))))))))
+i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
+
+theorem csuba_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1)))
+(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear
+x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to
+((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2)
+\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda
+(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n)
+(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4
+(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0
+(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2
+C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda
+(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear
+(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O
+c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1))
+\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b)
+t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst)
+u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c]))
+(CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead
+d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
+(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind
+Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst)
+u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0)
+\Rightarrow t0])) (CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B
+Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba
+g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0
+(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0:
+B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in (let H15 \def
+(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1
+H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind
+(ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda
+(d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst)
+u1))).(\lambda (H19: (csuba g d1 x1)).(or_introl (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2
+(CHead x1 (Bind Abst) u1) (CHead x1 (Bind Abst) u1) H18 (clear_bind Abst x1
+u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1
+(Bind Abbr) x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1
+u1 (asucc g x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))
+x1 x2 x3 (getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2)
+H18 (clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8))
+H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f)
+t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst)
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0
+(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
+c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1:
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2)
+\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda
+(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
+C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c:
+C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat
+f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1)
+(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def
+(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1)
+H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2))
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1)
+x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1
+H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
+(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3
+(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2
+(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in
+(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2
+(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
+A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba
+g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity
+g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13
+(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead
+x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15))
+H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1:
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2)
+\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat
+f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def
+(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v)))))
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat
+f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12:
+(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0
+(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2)
+x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2))
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4)
+x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5
+x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda
+(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda
+(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl
+(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7:
+C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6)
+x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c:
+C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13
+x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or
+(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
+(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
+n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7
+(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C
+(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23)
+H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23:
+(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1
+x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9
+x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6
+c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22))
+H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1
+H2)))) H0))))))).
+
+theorem csuba_getl_abst_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g
+c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u)))
+(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x:
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
+Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1
+(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))
+(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear
+(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u)
+n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0:
+C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u))
+\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4:
+(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0:
+K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind
+Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b:
+B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear
+(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u)
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in
+((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0
+in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
+Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u)
+t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda
+(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda
+(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
+(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst
+H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c
+(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u
+H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C
+(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1:
+C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19:
+(csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1
+(getl_intro i c2 (CHead x1 (Bind Abst) u) (CHead x1 (Bind Abst) u) H18
+(clear_bind Abst x1 u)) H19))))) H17)) (\lambda (H17: (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2:
+C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Void)
+x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 x2 (getl_intro i c2
+(CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 (clear_bind Void x1
+x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5:
+(drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f)
+t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c:
+C).((drop i O c (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c)
+\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop
+n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or
+(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n c2
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f)
+t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C
+x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1
+(CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind
+Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) f t) in (let H11
+\def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst)
+u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abst) u)))
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead
+d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (H12:
+(csuba g x2 (CHead d1 (Bind Abst) u))).(\lambda (H13: (clear c2 x2)).(let H_x
+\def (csuba_gen_abst_rev g d1 x2 u H12) in (let H14 \def H_x in (or_ind (ex2
+C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead
+d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda
+(d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
+(or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl
+O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
+d2 d1))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst)
+u))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c:
+C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u) H16) in (or_introl (ex2 C
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst)
+u) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3:
+C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind Void)
+x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c:
+C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in (or_intror (ex2 C
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3
+x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 (drop_refl c2) H18)
+H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8:
+((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2:
+C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9:
+(drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10:
+(csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in
+(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1
+(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
+T).(drop n O e (CHead x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S
+n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2:
+B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind
+x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def
+(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C
+(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2:
+C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: C).(\lambda (H15: (csuba
+g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def
+(csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
+x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8:
+T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g
+x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead
+x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C
+(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
+(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (H23: (getl
+n x7 (CHead x9 (Bind Abst) u))).(\lambda (H24: (csuba g x9 d1)).(or_introl
+(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl
+(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
+g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8
+H20 (CHead x9 (Bind Abst) u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl
+(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9:
+C).(\lambda (x10: T).(\lambda (H23: (getl n x7 (CHead x9 (Bind Void)
+x10))).(\lambda (H24: (csuba g x9 d1)).(or_intror (ex2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10
+(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24))))))
+H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4)))))))
+x H1 H2)))) H0))))))).
+
+theorem csuba_getl_abbr_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba
+g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1)))
+(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))
+(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead
+d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c
+(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3
+(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_:
+(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr)
+u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba
+g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0:
+C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1))
+\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1
+(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr)
+u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear
+(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2
+c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b:
+B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear
+(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u1)
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6))
+in ((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
+(_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow
+(match k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
+(Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b)
+t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
+Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr)
+u1) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1
+x0)).(\lambda (c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r
+T t (\lambda (t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let
+H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0)
+u1))) H13 Abbr H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O
+c1 (CHead c (Bind Abbr) u1))) H14 d1 H11) in (let H16 \def
+(csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in (or3_ind (ex2 C (\lambda
+(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C
+(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1
+(Bind Abbr) u1))).(\lambda (H19: (csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2
+(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1
+u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1
+(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1
+x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3
+(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18
+(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
+(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind
+Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead
+x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16))))))))))
+H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f)
+t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr)
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop
+n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3
+(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8:
+(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g
+c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead
+x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def
+(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind
+Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f)
+t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2:
+C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2))
+(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2:
+C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13:
+(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14
+\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda
+(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3:
+C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba
+g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead
+x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2
+(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
+A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba
+g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity
+g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13
+(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst)
+x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind
+Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2
+(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in
+(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3
+(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11))))))))
+(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1
+(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let
+H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T
+(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b)
+v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0
+(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3
+(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14
+\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind
+C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2:
+C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16:
+(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let
+H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S
+n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8:
+T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g
+x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead
+x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C
+(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S
+n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
+(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
+(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24:
+(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
+(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr)
+u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11:
+A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24:
+(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda
+(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20
+(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22:
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T
+(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
+T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23:
+(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9
+d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro
+C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10
+(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24))))))
+H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4)))))))
+x H1 H2)))) H0))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csuba/defs.ma".
+
+theorem csuba_refl:
+ \forall (g: G).(\forall (c: C).(csuba g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0))
+(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0
+c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/csuba.ma".
+
+theorem csubc_arity_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to
+(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
+c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t
+a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))).
+
+theorem csubc_arity_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to
+((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to
+(arity g c1 t a))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
+c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda
+(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2
+H) H0)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/fwd.ma".
+
+theorem csubc_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall
+(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda
+(e2: C).(csubc g e1 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1
+e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c
+c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0
+e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2:
+C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def
+(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2
+C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g
+e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
+(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq
+C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda
+(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3
+g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
+a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2))
+(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda
+(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
+C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2
+(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x
+(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda
+(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2:
+C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind
+b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind
+b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda
+(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind
+b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda
+(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7:
+(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2
+C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b)
+u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b]))
+(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda
+(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g
+(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0
+(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2))
+(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2
+H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda
+(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0)
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind
+Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e
+c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2))
+(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind
+x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B
+x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2)
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc
+g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0:
+K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
+(Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void
+(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2))
+(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda
+(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead
+e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2)
+(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1))))))))
+(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
+((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2))
+(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u:
+T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x
+\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind
+(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3:
+C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
+A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda
+(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
+(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3:
+C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2:
+C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C
+(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda
+(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
+C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f)
+u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda
+(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c
+e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda
+(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2:
+C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2)))
+(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c
+x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda
+(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5))))
+H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
+A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda
+(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
+(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2
+e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6:
+(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda
+(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C
+(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0
+e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f)
+(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in
+(False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2))
+(\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4:
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
+T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3:
+C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2:
+C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda
+(x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0)
+x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0
+Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2)
+(\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2:
+C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match
+ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat
+_) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2:
+C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9))
+c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/defs.ma".
+
+include "LambdaDelta-1/sc3/props.ma".
+
+theorem csubc_csuba:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba
+g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
+c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda
+(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
+(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda
+(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b:
+B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v:
+T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w:
+T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g
+c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sc3/defs.ma".
+
+inductive csubc (g: G): C \to (C \to Prop) \def
+| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
+| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
+(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
+| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
+(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g
+a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr)
+w))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/fwd.ma".
+
+include "LambdaDelta-1/sc3/props.ma".
+
+theorem csubc_drop_conf_O:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h
+O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2:
+C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1:
+C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2)
+\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1
+e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H:
+(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n)
+c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda
+(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1:
+(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O
+O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2
+e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c:
+C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c
+e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2:
+C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2))))
+(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1:
+C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2)
+\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1
+e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h:
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall
+(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2
+e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c
+k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind
+C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2))
+(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O
+c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1
+(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0:
+(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t)
+c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g
+e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2:
+C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l
+g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C
+c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T
+(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b)
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3)))))
+(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1
+e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t)))
+(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2
+(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2:
+C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x:
+C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c
+x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop
+(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k
+n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C
+(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2))
+(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2:
+C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x
+x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n)
+O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0
+H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c
+t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2
+C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k
+(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda
+(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3
+g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C
+(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2))))
+(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1))
+(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k
+(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g
+(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda
+(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r
+(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2:
+C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2:
+C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1
+e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g
+e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1)
+e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13
+x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2)))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void)))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
+(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c
+c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2
+e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda
+(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8:
+(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C
+(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2))))
+(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1))
+(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k
+(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g
+(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda
+(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r
+(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2:
+C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2:
+C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1
+e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g
+e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2)
+e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2)
+H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)).
+
+theorem drop_csubc_trans:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
+C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
+(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
+(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
+C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g
+(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def
+(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1))
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c
+c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
+e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h
+n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h:
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
+(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
+(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t)
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
+O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1)
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
+(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2
+e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
+(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k
+t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t)))))
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t)
+c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
+C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
+(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to
+(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
+e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1)
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k
+x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1:
+C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r
+T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n)
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def
+(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C
+(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1
+(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
+(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
+a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n)
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))
+(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda
+(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k
+x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n)
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))
+(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12:
+(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda
+(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r
+k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1:
+C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1)))
+(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2:
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c
+x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
+(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0
+x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))
+(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g
+(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3:
+T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C
+e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14:
+(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C
+(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S
+n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))))
+(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n
+(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3:
+C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
+e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1))))))))
+H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r
+k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0:
+K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
+(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0
+\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind
+C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C
+(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1:
+C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1)))
+(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c
+x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr)
+x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst)
+n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19
+Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g
+(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g
+x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda
+(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
+T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3:
+C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1:
+C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n)
+x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11:
+(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind
+Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0
+x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda
+(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r
+k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0:
+nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to
+(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1:
+C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n)
+x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda
+(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind
+Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3
+(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1))
+c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc
+g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4)))
+(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1))
+c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g
+c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2)
+x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void)
+n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18
+x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n
+x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4)))))))))
+(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+
+theorem csubc_drop_conf_rev:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
+C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
+(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
+(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
+C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1
+(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def
+(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n)))
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1
+c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
+e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h
+n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h:
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
+(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
+(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t)
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
+O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1)
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
+(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1
+e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
+(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c
+k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t)))))
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k
+t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
+C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
+(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to
+(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
+e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1)
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0
+k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc
+g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h
+(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1))
+(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def
+(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
+x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1
+(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g
+(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a
+x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq
+C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_:
+T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
+x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1:
+C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x:
+C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x
+x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1:
+C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k
+n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in
+(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g
+c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2:
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2
+c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1:
+C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v)))))
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2
+C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead
+c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2
+(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g
+(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2
+(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1
+c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let
+H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c
+k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3
+(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda
+(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr)
+H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5
+(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda
+(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc
+g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind
+Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1:
+C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1:
+C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1
+(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x:
+C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C
+(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1:
+C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x
+(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst
+g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19)
+(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n)
+H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T
+(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
+(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1
+x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1:
+T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1:
+C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n)
+c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))
+(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1
+(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13:
+(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3
+(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1
+c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let
+H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c
+k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3
+(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda
+(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2)
+H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5
+(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1:
+C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1
+(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h
+H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1
+x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n)
+c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind
+x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n
+x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S
+n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind
+x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4))
+(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n
+x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10))
+H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/drop.ma".
+
+theorem drop1_csubc_trans:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2
+e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
+c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def
+(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c:
+C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1
+e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2:
+C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1)
+\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
+c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n
+n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def
+(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
+(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda
+(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))
+(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x
+e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C
+(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2
+c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g
+x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def
+H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1:
+C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1))
+(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0
+x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0
+n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
+
+theorem csubc_drop1_conf_rev:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1
+e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
+c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def
+(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c:
+C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1
+e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2:
+C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2)
+\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
+c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n
+n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def
+(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
+(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda
+(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))
+(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x
+e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C
+(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1
+c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g
+x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8
+\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1:
+C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1))
+(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0
+x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0
+n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/defs.ma".
+
+theorem csubc_gen_sort_l:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g
+(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c))))
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0)
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C
+c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v)
+(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
+_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v)
+(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
+(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
+c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort
+n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead
+c2 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1
+(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_:
+(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2
+w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def
+(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
+\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr)
+w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))).
+
+theorem csubc_gen_head_l:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k:
+K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x
+(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2:
+C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w)))))
+(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T
+(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b)
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1
+c2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k:
+K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v)
+(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2:
+C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))
+(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead
+c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k
+(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1
+c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda
+(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda
+(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))
+(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0
+(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_:
+T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n)
+(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _
+_ _) \Rightarrow False])) I (CHead c1 k v) H1) in (False_ind (or3 (ex2 C
+(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g
+c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort
+n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
+(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
+a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2:
+T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2:
+C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v))
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3:
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda
+(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0:
+K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k
+v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
+(CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K (\lambda
+(e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0
+| (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in
+((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead
+c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq
+C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C
+(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead
+c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc
+g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g
+a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3
+w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
+(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3:
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3
+(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k
+(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
+c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v))
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3:
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda
+(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let
+H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in
+(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v)))
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda
+(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind
+b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
+c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v)))
+(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0
+H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda
+(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2
+C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2
+(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
+(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
+a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not
+(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead
+c0 (Bind Void) u1) (CHead c1 k v))).(let H5 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
+(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4)
+in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _)
+\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in ((let H7
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0
+(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void)
+k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c:
+C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead
+c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T
+(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind
+b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
+c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c
+c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1
+(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v)))
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda
+(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0)
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind
+Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
+c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v)))
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T
+(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b)
+u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_:
+T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
+T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2
+(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3)))
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind
+Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C
+(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda
+(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda
+(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0)
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void)
+(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B
+b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
+c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda
+(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C
+(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6))
+H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0
+c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3:
+C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
+T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3:
+(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2
+w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0
+(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind
+Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1
+k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow
+t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K
+(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0
+(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C
+c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def
+(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C
+(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3)))
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind
+Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead
+c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g
+(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3
+g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind
+C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K
+k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3:
+C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
+Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0)
+c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3
+w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
+c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
+T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst)
+(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w)
+(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3
+(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g
+c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g
+a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3
+w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
+(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3:
+C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3:
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0:
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr)
+w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v))))
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))))
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead
+c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda
+(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w)
+(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g
+(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3
+g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2
+(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0)))
+H)))))).
+
+theorem csubc_gen_sort_r:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x
+(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0))))
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0)
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C
+c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v)
+(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
+_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v)
+(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
+(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1
+c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort
+n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
+_ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1
+(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2
+(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_:
+(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2
+w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def
+(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
+\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst)
+v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))).
+
+theorem csubc_gen_head_r:
+ \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k:
+K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x
+(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1:
+C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v)))))
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T
+(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
+(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k:
+K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w)
+(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1:
+C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))
+(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead
+c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K
+k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1
+c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda
+(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda
+(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))
+(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead
+c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K
+k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1
+c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k
+w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda
+(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2)))
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind
+Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n)
+(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g
+(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a
+c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq
+C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1:
+C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0:
+C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w))
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_:
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v:
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0:
+K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(let
+H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0
+v) (CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e
+in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1
+_) \Rightarrow k1])) (CHead c0 k0 v) (CHead c2 k w) H3) in ((let H6 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v)
+(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0
+c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead
+c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t)
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3
+g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3
+g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1:
+T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k
+(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3
+k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind
+Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3
+c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a)
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead
+c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda
+(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1
+(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda
+(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
+(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2
+H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8)
+in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w)))
+(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0:
+T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda
+(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w)
+(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
+T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not
+(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g
+c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k
+w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w))
+H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0:
+C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w))
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_:
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v:
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b:
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b)
+u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e
+in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead
+_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let
+H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0
+(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda
+(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead
+c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda
+(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
+T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3:
+C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda
+(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void)
+v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
+b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3
+c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g
+c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2
+(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w)))
+(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
+T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3:
+C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda
+(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void)
+v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind
+b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3
+c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2
+C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda
+(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst)
+v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
+(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind
+Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3:
+C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst)
+v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
+(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind
+Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_:
+C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_:
+B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead
+c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K
+(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not
+(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g
+c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K
+(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda
+(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k
+w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_:
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v:
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v:
+T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0:
+T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr)
+w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
+\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0)
+(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq
+C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8)
+in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in
+(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3
+(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g
+c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3
+g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0:
+A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda
+(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind
+C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K
+k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3:
+C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr)))))
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind
+Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3
+c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0)
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2
+w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C
+c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
+T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not
+(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g
+c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0:
+K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0
+w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda
+(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda
+(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst)
+v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3
+v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead
+c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3:
+C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0:
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst)
+v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3
+v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead
+c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda
+(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v)
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3
+g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0:
+A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead
+c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0)))
+H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/drop.ma".
+
+include "LambdaDelta-1/csubc/clear.ma".
+
+theorem csubc_getl_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i
+c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2:
+C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda
+(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1
+\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e))
+(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2))
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1
+x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2
+c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2))
+(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2))
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O
+c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1
+H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2))
+(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2))
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0
+x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2
+e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8)
+H9)))) H7)))))) H4)))))) H1)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/defs.ma".
+
+include "LambdaDelta-1/sc3/props.ma".
+
+theorem csubc_refl:
+ \forall (g: G).(\forall (c: C).(csubc g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0))
+(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0
+c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst0/props.ma".
+
+include "LambdaDelta-1/csubst0/fwd.ma".
+
+include "LambdaDelta-1/clear/fwd.ma".
+
+theorem csubst0_clear_O:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
+(\forall (c: C).((clear c1 c) \to (clear c2 c))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2
+c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
+(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n)
+c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda
+(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0:
+C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t)
+c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
+T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0:
+T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C
+c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k
+x0) (\lambda (c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c
+k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0))))
+(\lambda (b: B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
+(eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
+| (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind
+b) x0) c0) H8))))) (\lambda (f: F).(\lambda (H6: (clear (CHead c (Flat f) t)
+c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1
+(\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0
+(clear_gen_flat f c c0 t H6) f x0))))) k H1 H3) c2 H4)))))) H2)) (\lambda
+(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))
+(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq
+nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5:
+(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3
+c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0
+x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear
+(CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8
+\def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7)
+in (False_ind (clear (CHead x0 (Bind b) t) c0) H8))))) (\lambda (f:
+F).(\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s
+(Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c
+x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t
+H6)) f t))))) k H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda
+(_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
+nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k
+x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c
+x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) (K_ind
+(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to
+(clear (CHead x1 k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear (CHead c
+(Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def
+(eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8)
+in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9))))) (\lambda (f:
+F).(\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s
+(Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
+x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v
+t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t
+H7)) f x0)))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O
+H0))))))))))) c1).
+
+theorem csubst0_clear_O_back:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
+(\forall (c: C).((clear c2 c) \to (clear c1 c))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c
+c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
+(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2
+c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c:
+C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to
+(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O
+v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda
+(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear
+(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O
+(s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v
+t x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead c
+k x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead
+c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7:
+(eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0)
+c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda
+(f: F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead
+c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n
+v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t)))))
+k H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0)
+(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k
+x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c
+x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k
+t) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0
+k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: (eq
+nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let
+H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7)
+in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda (f:
+F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0
+(Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v
+c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t
+H8)) f t))))) k H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda
+(x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1
+k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c
+x1)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x1 k
+x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead
+x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8:
+(eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0)
+c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda
+(f: F).(\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead
+x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n:
+nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda
+(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0
+(clear_gen_flat f x1 c0 x0 H9)) f t)))))) k H3 H7))))))))) H2))
+(csubst0_gen_head k c c2 t v O H0))))))))))) c1).
+
+theorem csubst0_clear_S:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0
+(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4
+B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq
+C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2))))))))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c
+c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c:
+C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4
+(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))))))))
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i:
+nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4
+(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t)
+c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
+T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda
+(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
+(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k
+x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda
+(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat
+(S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b)
+u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))
+(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
+(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e:
+nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S
+n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1
+(\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind
+b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T
+(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
+(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b)
+x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda
+(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq
+C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead
+e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b)
+t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b)
+x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
+t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
+(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2
+(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1))))))
+(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0
+(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0
+(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear
+(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let
+H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in
+(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i)
+H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))
+(clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))) k H1 H3) c2
+H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2
+(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1:
+nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0
+k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda
+(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat
+(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b)
+u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))
+(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
+(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e:
+nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S
+n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1
+(\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind
+b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T
+(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
+(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b)
+t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
+(CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2
+(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t))
+(ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b)
+t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
+t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
+(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2
+(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
+(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t
+(refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0
+(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear
+(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let
+H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in
+(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i)
+H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in
+(or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat
+f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
+t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t)))
+(\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t)
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3:
+C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind
+x2) x4))).(\lambda (H13: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H14:
+(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
+t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
+t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0
+(CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
+t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0
+(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2
+(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t)
+H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2)
+x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14:
+(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear
+(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0
+(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))) k H1 H3)
+c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda
+(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda
+(H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda
+(H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4
+(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind
+(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to
+(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda
+(H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b)
+x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
+(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i)
+(S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
+x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v
+t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4
+(clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1))))))
+(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
+T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
+(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1)))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3
+(clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda
+(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind
+b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0)
+u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0)
+u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))
+(ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
+(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2
+(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0)
+H11 H10)) c0 (clear_gen_bind b c c0 t H7)))))))) (\lambda (f: F).(\lambda
+(H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f)
+x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f)
+x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
+x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0
+n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c
+c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat
+f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f)
+x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0)))
+(\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0)
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4:
+C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind
+x3) x5))).(\lambda (H15: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H16:
+(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f)
+x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f)
+x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1
+(CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B
+C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
+c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f)
+x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
+T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1
+(CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2
+(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0)
+H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
+T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3)
+x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16:
+(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear
+(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1
+(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1
+H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1).
+
+theorem csubst0_clear_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0
+i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C
+(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or
+(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1:
+C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1
+u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2)
+e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear
+(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2))
+(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda
+(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2)
+(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1:
+C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind
+b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2))
+(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2)))
+(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda
+(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear
+(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1
+u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda
+(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear
+(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
+(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2
+(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k:
+K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
+T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2:
+C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0
+v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda
+(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0:
+K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C
+(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead
+c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u)
+e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3
+(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c))
+(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear
+(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0
+(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind
+b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4
+(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3
+(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2
+(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead
+c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in
+(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0
+v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u)
+e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear
+(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear
+(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
+(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f
+u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda
+(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
+(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C
+(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
+(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x
+e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2)
+(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead
+c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
+(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f
+u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0
+i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0
+c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2)
+(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3
+e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind
+(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1)
+e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1:
+C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear
+(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c:
+C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s
+(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1)))))
+(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C
+(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1:
+C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0
+(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind
+b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4
+H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f:
+F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2
+(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2)
+(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3
+e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0
+i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda
+(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C
+(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
+(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda
+(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C
+(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or
+(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1
+e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x:
+C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror
+(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1
+e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C
+(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
+(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k
+H3))))))))))))) i v c1 c2 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+include "LambdaDelta-1/C/defs.ma".
+
+inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
+| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1:
+T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i)
+v (CHead c k u1) (CHead c k u2))))))))
+| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s
+k i) v (CHead c1 k u) (CHead c2 k u))))))))
+| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall
+(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall
+(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2
+k u2)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst0/fwd.ma".
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+include "LambdaDelta-1/s/props.ma".
+
+theorem csubst0_drop_gt:
+ \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
+c1 e) \to (drop n O c2 e)))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
+\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda
+(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
+O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H:
+((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
+(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop
+n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
+(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
+c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))
+(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v
+(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1)
+e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0)
+O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0)
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop
+(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee
+in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6))
+e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda
+(H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i
+v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k
+t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S
+n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k
+x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t
+x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let
+H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
+T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop
+(S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda
+(n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop
+(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1)
+v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
+e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0)
+e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda
+(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
+(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
+e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c
+e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c
+e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1)
+v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
+e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O)
+(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
+(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1
+O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m:
+nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
+(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13))
+(lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2
+H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda
+(x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6:
+(eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C
+(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
+c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
+e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1:
+nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0
+n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c
+c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
+e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t)
+e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda
+(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
+(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
+e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0
+x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f:
+F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3:
+C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt
+(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq
+nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat
+f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7
+e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e)
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
+n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS
+x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4))
+(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0:
+T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k
+x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
+x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda
+(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1:
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
+(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5)
+in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
+H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3:
+C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop
+(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to
+(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop
+(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0:
+T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c
+e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S
+n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11)
+x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda
+(H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3)
+\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
+e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O)
+(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
+(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2
+O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2
+nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m
+n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m:
+nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x:
+nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat
+f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k
+(drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head
+k c c2 t v i H2))))))))))) c1)))))) n).
+
+theorem csubst0_drop_gt_back:
+ \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
+c2 e) \to (drop n O c1 e)))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
+\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda
+(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
+O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H:
+((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
+(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop
+n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
+(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
+c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))
+(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i
+v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2
+e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e))))))))
+(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v
+c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c
+e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v:
+T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
+(H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e)
+(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k
+x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda
+(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
+c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c
+e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1:
+nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall
+(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1)
+(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e)))))
+(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
+(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
+(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda
+(H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t)))))
+(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
+(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
+(S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda
+(H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O
+(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c
+e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e)
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
+n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k
+H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S
+n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq
+nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7:
+(csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0)
+O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1:
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
+(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5)
+in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1)
+H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
+(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
+n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to
+(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt
+(s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0
+e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13)
+t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0:
+T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
+e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S
+n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O)
+(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
+(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop
+(Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m:
+nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
+(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13)
+t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0
+H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O
+(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
+nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k
+x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c
+x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3
+(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1:
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
+(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5)
+in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
+H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
+(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
+n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to
+(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt
+(s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1
+e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14)
+t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0:
+T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
+e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S
+n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O)
+(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
+(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop
+(Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m:
+nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
+(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
+(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14)
+t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0
+H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
+
+theorem csubst0_drop_lt:
+ \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
+c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k
+w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
+n)) v e1 e2))))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
+\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T
+T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
+O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i:
+nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v:
+T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O
+c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0:
+nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k:
+K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r
+nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O))
+(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1))
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1)
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
+k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1)
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C
+(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k:
+K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
+T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0
+O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0
+O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda
+(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0))
+in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s
+k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead
+e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0
+(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in
+(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k
+u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k
+i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
+u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
+v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0
+u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k
+i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
+u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
+v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k
+u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3
+c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0
+O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1
+e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1
+u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r
+nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O))
+(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1))
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k
+u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
+k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1)
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k
+u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u
+w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4
+u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5
+H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda
+(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1:
+C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e:
+C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))
+(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead
+e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H:
+(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c
+e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1:
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1)
+c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind
+(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S
+n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq
+nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c:
+C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind
+nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop)
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind
+(or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort
+n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1
+k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
+(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C
+T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
+(H2: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
+n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))
+(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0
+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4:
+(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_:
+(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
+n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
+H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
+T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
+(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
+(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
+T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
+(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
+(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i
+(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1)
+(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
+(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
+(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
+(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
+(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
+(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
+u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
+e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c
+k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
+k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
+(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b)
+x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
+e2))))))) (drop_drop (Bind b) n0 c e H9 x0)))))) (\lambda (f: F).(\lambda
+(H9: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall
+(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0)
+O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
+e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
+x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2))))))) (drop_drop (Flat f) n0 c e H9 x0)))))) k (drop_gen_drop k c e t n0
+H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4:
+(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6:
+(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop
+(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
+H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
+T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
+(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
+(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
+T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
+(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
+(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i
+(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1)
+(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
+(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
+(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
+(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
+(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
+(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
+u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
+e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0
+k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
+k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
+(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b)
+x1))).(let H12 \def (IHn x1 (le_S_n (S n0) x1 H11) c x0 v H6 e H9) in
+(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
+n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4
+(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H13: (drop n0 O x0
+e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
+e2))))))) (drop_drop (Bind b) n0 x0 e H13 t))) (\lambda (H13: (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
+n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0
+(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x4))).(\lambda (H15:
+(drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H16: (subst0 (minus x1 (s x2 n0))
+v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
+(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
+(CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4)
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
+x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4))
+(drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H15 t) (eq_ind_r nat (S (s x2 n0))
+(\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H16 (s x2 (S
+n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
+n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
+k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x0
+(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
+C).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
+(drop n0 O x0 (CHead x4 x2 x5))).(\lambda (H16: (csubst0 (minus x1 (s x2 n0))
+v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
+(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
+(CHead x0 (Bind b) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
+x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
+x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H15 t) (eq_ind_r nat (S (s x2
+n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H16 (s
+x2 (S n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex4_5 K C C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2
+k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
+n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T
+T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
+e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
+(drop n0 O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2 n0))
+v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 n0)) v x3 x4)).(eq_ind_r C
+(CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t)
+c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
+(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x0 (Bind b) t)
+(CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e0 k0 u)))))) (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
+(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
+(CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
+e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) x2 x3 x4 x5 x6
+(refl_equal C (CHead x3 x2 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x6)
+H15 t) (eq_ind_r nat (S (s x2 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind
+b) x1) n1) v x5 x6)) H16 (s x2 (S n0)) (s_S x2 n0)) (eq_ind_r nat (S (s x2
+n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H17 (s
+x2 (S n0)) (s_S x2 n0)))) e H14)))))))))) H13)) H12)))))) (\lambda (f:
+F).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(\lambda (H10: ((\forall (c3:
+C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
+x1))).(let H12 \def (H10 x0 v H6 e H9) in (or4_ind (drop (S n0) O x0 e)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1
+(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2)))))))) (\lambda (H13: (drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O
+(CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x0 e H13
+t))) (\lambda (H13: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x1 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S n0))) v u
+w))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Flat f) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3:
+C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2
+x4))).(\lambda (H15: (drop (S n0) O x0 (CHead x3 x2 x5))).(\lambda (H16:
+(subst0 (minus x1 (s x2 (S n0))) v x4 x5)).(eq_ind_r C (CHead x3 x2 x4)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
+k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 x2
+x4)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x3 x2 x4) (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (ex3_4_intro K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))
+x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead
+x3 x2 x5) H15 t) H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
+(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead
+e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
+x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
+C).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
+(drop (S n0) O x0 (CHead x4 x2 x5))).(\lambda (H16: (csubst0 (minus x1 (s x2
+(S n0))) v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop
+(S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
+(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
+(CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S
+n0) O (CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
+x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
+x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H15 t) H16)) e H14))))))))
+H13)) (\lambda (H13: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
+(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
+(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K
+C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
+(drop (S n0) O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2
+(S n0))) v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 (S n0))) v x3
+x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead
+x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O
+(CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
+x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
+e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f)
+n0 x0 (CHead x4 x2 x6) H15 t) H16 H17)) e H14)))))))))) H13)) H12)))))) k
+(drop_gen_drop k c e t n0 H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3:
+(ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s
+k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
+c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
+nat).(\lambda (H4: (eq nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k
+x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c
+x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
+(S n0))) v e1 e2))))))))) (let H8 \def (eq_ind nat i (\lambda (n1:
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S
+n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
+e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1
+(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H4) in (let H9 \def (eq_ind
+nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k
+x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
+(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
+(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
+(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2)
+(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2))))))
+(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0
+u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1
+e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1
+k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0)
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4
+K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
+k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
+(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b)
+x2))).(let H13 \def (IHn x2 (le_S_n (S n0) x2 H12) c x1 v H7 e H10) in
+(or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
+n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4
+(drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x1
+e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
+e2))))))) (drop_drop (Bind b) n0 x1 e H14 x0))) (\lambda (H14: (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
+n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1
+(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x5))).(\lambda (H16:
+(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H17: (subst0 (minus x2 (s x3 n0))
+v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
+(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
+(CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5)
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
+x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x2) (s k0 (S n0))) v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5))
+(drop_drop (Bind b) n0 x1 (CHead x4 x3 x6) H16 x0) (eq_ind_r nat (S (s x3
+n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H17 (s
+x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
+n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
+k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1
+(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
+C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
+(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 n0))
+v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
+(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
+(CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
+x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3
+x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H16 x0) (eq_ind_r nat (S (s
+x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H17
+(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead
+e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0
+n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T
+T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
+e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
+T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
+(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 n0))
+v x6 x7)).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C
+(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0)
+c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
+(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
+(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
+(CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
+e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
+x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3
+x7) H16 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s
+(Bind b) x2) n1) v x6 x7)) H17 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s
+x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18
+(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f:
+F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3:
+C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0:
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
+x2))).(let H13 \def (H11 x1 v H7 e H10) in (or4_ind (drop (S n0) O x1 e)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2
+(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e)
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2)))))))) (\lambda (H14: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O
+(CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Flat f) x2) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H14
+x0))) (\lambda (H14: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
+e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus x2 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
+w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
+C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3
+x5))).(\lambda (H16: (drop (S n0) O x1 (CHead x4 x3 x6))).(\lambda (H17:
+(subst0 (minus x2 (s x3 (S n0))) v x5 x6)).(eq_ind_r C (CHead x4 x3 x5)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T
+T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
+k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
+x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
+(ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
+w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1
+(CHead x4 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C
+C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
+(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead
+e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
+x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
+(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
+C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
+(drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3
+(S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop
+(S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0
+(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
+(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))))
+(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K
+C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
+(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1
+(CHead x5 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C
+C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
+w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1
+e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4
+(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
+C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e
+(CHead x4 x3 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 x3
+x7))).(\lambda (H17: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda
+(H18: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3
+x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K
+C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
+k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
+x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
+(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
+x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3
+x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))) k (drop_gen_drop k c e
+t n0 H2) H8 H9) i H4))) c2 H5)))))))) H3)) (csubst0_gen_head k c c2 t v i
+H1))))))))))) c1)))))) n).
+
+theorem csubst0_drop_eq:
+ \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
+n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
+C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1
+e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
+c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1
+e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
+(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0
+y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c:
+C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0
+t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0:
+K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2:
+T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4
+(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
+c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f)
+u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
+k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1)
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2)
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda
+(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
+u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat
+(S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with
+[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4
+(drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
+c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2)
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
+u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind
+b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b)
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
+i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind
+nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O
+(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f)
+u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0
+(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
+(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f)
+u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat
+f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
+C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f)
+u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1))
+(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
+C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
+i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall
+(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u)
+(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0
+i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0
+i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f)
+u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
+e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
+(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u)
+(CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0
+(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S
+i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i)
+(S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1
+(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1
+e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3
+c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i
+O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
+(drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0
+(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0
+n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
+nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
+nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat
+f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3
+(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C
+T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u)
+(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4
+(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u))
+(drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
+T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
+C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
+(s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0
+u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat
+f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f)
+u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
+k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1)
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2)
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
+\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
+in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind
+b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i)
+(S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead
+c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2))))))))
+H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda
+(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq
+nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i
+c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T
+T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2
+(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
+e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda
+(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u
+w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def
+(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8
+\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in
+(eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2)
+(CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0
+u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F
+C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3
+(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f)
+u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
+u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
+u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1))
+(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
+H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
+((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
+(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda
+(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall
+(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort
+n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1)
+e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop
+(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0)
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4
+(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in
+nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1))
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f)
+u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c:
+C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2)
+\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda
+(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v
+(CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t)
+e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s
+k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2))))
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda
+(_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda
+(_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
+n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq
+nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k
+u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat
+(S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5:
+(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
+n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
+n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
+k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c
+e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
+\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def
+(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H8) in
+(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c
+(Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H6 x0)))))))
+(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq
+nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0:
+nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda
+(n1: nat).(subst0 n1 v t x0)) H5 (S n0) H8) in (or4_intro0 (drop (S n0) O
+(CHead c (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) x0)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (drop_drop (Flat f) n0 c e H6 x0))))))) k (drop_gen_drop k c
+e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0)
+(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t))))
+(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O
+c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0)
+(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1
+v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
+n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c
+e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
+\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def
+(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H8) in (let
+H10 \def (IHn c x0 v H9 e H6) in (or4_ind (drop n0 O x0 e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11:
+(drop n0 O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x0 e H11
+t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
+x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x0 (Bind
+b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop n0 O
+x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 x5)).(eq_ind_r C
+(CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind
+b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
+x2) x4)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x3 (Flat x2) x4) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
+x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4))
+(drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) e H12))))))))
+H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
+x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0
+(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop n0 O
+x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 x4)).(eq_ind_r C
+(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind
+b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
+x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2)
+x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
+(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H13 t) H14)) e H12))))))))
+H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
+O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
+(H13: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v x5
+x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
+(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
+x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0
+x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10)))))))
+(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq
+nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0:
+nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda
+(n1: nat).(csubst0 n1 v c x0)) H5 (S n0) H8) in (let H10 \def (H x0 v H9 e
+H6) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11:
+(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat
+f) n0 x0 e H11 t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
+O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda
+(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop
+(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4
+x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
+(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3
+(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14))
+e H12)))))))) H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
+C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
+(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop
+(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3
+x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
+(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
+C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
+(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead
+x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H13 t)
+H14)) e H12)))))))) H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0
+(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
+(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
+f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
+(H13: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v
+x5 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2)
+x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
+(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
+x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat
+f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
+f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
+f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0
+x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10)))))))
+k (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
+(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
+C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F
+C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4:
+(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6:
+(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop
+(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
+n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c
+e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(let H9 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
+\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H10 \def
+(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H9) in (let
+H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in
+(let H12 \def (IHn c x1 v H10 e H7) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T
+T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0)
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13:
+(drop n0 O x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H13
+x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
+x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind
+b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop n0 O
+x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 x6)).(eq_ind_r C
+(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
+b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
+x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
+x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5))
+(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) H16)) e H14))))))))
+H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
+x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1
+(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
+T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H15: (drop n0 O
+x1 (CHead x5 (Flat x3) x6))).(\lambda (H16: (csubst0 O v x4 x5)).(eq_ind_r C
+(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
+b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
+x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
+x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6))
+(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14))))))))
+H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
+O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
+T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda
+(H15: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H16: (subst0 O v x6
+x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T
+T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
+(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
+x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0
+x1 (CHead x5 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13))
+H12)))))))) (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c
+e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(let H9 \def (f_equal nat
+nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H8) in (let H10 \def
+(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 (S n0) H9) in
+(let H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S
+n0) H9) in (let H12 \def (H x1 v H10 e H7) in (or4_ind (drop (S n0) O x1 e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: (drop (S n0) O
+x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H13
+x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
+O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda
+(x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop
+(S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5
+x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead
+x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0)
+u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4
+(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H15 x0)
+H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind
+F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4:
+C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat
+x3) x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda
+(H16: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat
+f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0)
+u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1
+(CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) H13)) (\lambda (H13:
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda
+(x6: T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3)
+x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda
+(H16: (subst0 O v x6 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C
+(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat
+f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
+f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat
+x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
+(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7
+(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5
+(Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) H12)))))))) k
+(drop_gen_drop k c e t n0 H1) H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2
+t v (S n0) H0))))))))))) c1)))) n).
+
+theorem csubst0_drop_eq_back:
+ \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
+n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
+C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2
+e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O
+c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2
+e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O
+O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
+(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda
+(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t:
+T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0
+t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda
+(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall
+(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s
+k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2))
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4:
+T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
+u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3:
+T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2)
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0
+u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
+u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
+i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def
+(eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in
+(False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b)
+u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead
+c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C
+C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
+(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b)
+u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq
+C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i)
+(CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
+u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f:
+F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i
+O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O
+H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f)
+u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0
+(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
+T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0)
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
+T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead
+e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c
+(Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
+(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C
+C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
+(CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1)
+(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
+(CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c
+(Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f)
+u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0)
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
+T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2))
+(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
+C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
+i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to
+(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead
+c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f)
+u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s
+k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0
+i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
+e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
+(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u)
+(CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S
+i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f)
+u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S
+i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead
+e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u)
+(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i:
+nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2:
+(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i
+v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
+e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def
+(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0
+n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
+C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
+nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
+nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
+(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
+u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f)
+u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4
+(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O
+(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C
+C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C
+(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u)
+(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
+(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3
+(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
+u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u))
+(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
+T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
+C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
+C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
+(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
+(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
+u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0
+(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
+(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2)
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat
+f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
+u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0
+u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
+(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3:
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
+(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
+\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
+in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind
+b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i)
+(S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
+u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i)
+(S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2
+(Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1
+(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3
+c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
+(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0)
+u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3:
+T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
+i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i
+O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
+(drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0
+(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
+(u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0
+c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
+(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
+u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3:
+T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
+e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0:
+nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda
+(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0:
+nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F
+C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
+C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3
+(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1)
+(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0
+n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
+n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O
+(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
+u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0)
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
+T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead
+e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
+u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1)
+(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2))
+(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
+H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
+((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
+(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind
+(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to
+(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort
+n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort
+c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat
+f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S
+n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O
+c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e:
+C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
+n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
+(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4
+(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda
+(x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2
+(CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2
+(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind
+(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to
+(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b:
+B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r
+(Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match
+e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1)
+\Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1
+(\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0)
+O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead
+e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f:
+F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r
+(Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S
+n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v
+t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
+(Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2))
+(\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0)
+(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t))))
+(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k
+t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0)
+(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1
+v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1
+(CHead x0 k t) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to
+((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
+C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
+e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b)
+x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
+\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def
+(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let
+H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
+(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
+x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15:
+(subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4
+(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c
+(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq
+C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2)
+x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
+(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5
+(refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3
+(Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O
+c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16:
+(csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0
+c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))))
+(\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8:
+(drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0:
+nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1:
+nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in
+(or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead
+e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0)
+O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e
+H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
+x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda
+(H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat
+f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
+(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13))))))))
+H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0)
+O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
+(x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop
+(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3
+x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4
+(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0)
+u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0)
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c
+(CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
+(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4
+(Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2)
+x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3
+x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4
+(Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0)
+u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0)
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6))
+(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e
+H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6))))))))
+H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda
+(j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0)
+O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda
+(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2
+(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2
+v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1
+(CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to
+((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
+C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
+e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b)
+x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
+\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def
+(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let
+H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10)
+in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
+(Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3)
+x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17:
+(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4
+(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c
+(CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14:
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq
+C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3)
+x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
+(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
+x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6
+(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4
+(Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
+T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O
+c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18:
+(csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0
+c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13))))))))
+(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9:
+(drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0:
+nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1:
+nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2
+(\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1
+v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
+(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
+e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda
+(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4
+(Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3)
+x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O
+(CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
+x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat
+f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead
+x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t)
+H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
+C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
+(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3)
+x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda
+(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat
+f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop
+(Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14))
+(\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0)
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C
+C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
+T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S
+n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda
+(H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat
+f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
+x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat
+x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e
+H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7))))))))))
+H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n).
+
+theorem csubst0_drop_lt_back:
+ \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O
+c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n)
+v e1 e2)) (\lambda (e1: C).(drop n O c1 e1))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
+\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
+(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
+c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda
+(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1
+c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0:
+nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2))
+(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C
+(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1)))
+(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O
+c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl
+c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt
+n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1
+c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
+(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
+c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda
+(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
+c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c
+e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1:
+C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda
+(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O
+(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2))
+(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
+(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C
+(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop
+(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
+C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda
+(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda
+(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0)
+O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1
+e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3:
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda
+(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead
+c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s
+k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t
+x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2
+(CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall
+(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S
+n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0
+(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))
+H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0)
+n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S
+n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v
+e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda
+(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to
+(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
+(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1:
+C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0
+n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0)
+O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3:
+C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3:
+C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
+C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
+(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda
+(H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind
+b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda
+(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2
+H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0:
+T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
+e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
+(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c
+e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop
+(r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2)
+(ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1:
+C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12
+t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda
+(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))
+(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead
+c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s
+k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c
+x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2
+(CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall
+(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S
+n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0
+(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))
+H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0)
+n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S
+n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v
+e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda
+(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to
+(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
+(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1:
+C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0
+n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0)
+O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3:
+C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3:
+C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
+C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
+(S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda
+(H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1
+H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2
+C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0
+O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
+(Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O
+(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1
+e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop
+(Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0
+(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C
+(\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
+c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
+(Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x
+e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind
+b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda
+(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1:
+C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
+(Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14))
+H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0:
+T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
+e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
+(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c
+e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop
+(r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def
+H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus
+x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0)
+O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0))
+v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))))
+(\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat
+f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2))
+(\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat
+f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1
+(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C
+(\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop
+(S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda
+(e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
+(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1
+(S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0)
+O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0))
+v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))
+(ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda
+(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0
+c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i
+H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0)
+O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1
+e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0:
+T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k
+x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
+x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda
+(c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
+c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3)
+(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1:
+C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat
+i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2)
+(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
+(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall
+(v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
+e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
+k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to
+((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O
+(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0))
+v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda
+(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b)
+x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0)
+O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1
+e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S
+n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x
+\def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in
+(or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1
+e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b)
+t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1:
+C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c
+e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
+(Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C
+(\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
+c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2))
+(\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t)
+e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1:
+C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16:
+(csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror
+(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0
+(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t)
+e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda
+(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0
+c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall
+(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3:
+C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
+C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
+(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda
+(H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in
+(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c
+e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
+(CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl
+(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0
+(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f)
+t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda
+(e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
+c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2))
+(\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f)
+t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda
+(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda
+(H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c
+x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
+(CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2
+(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x
+H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10
+(drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c
+c2 t v i H1))))))))))) c1)))))) n).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst0/defs.ma".
+
+theorem csubst0_gen_sort:
+ \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0
+i v (CSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n)
+(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y:
+C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda
+(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P)))))
+(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq
+C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in
+(False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1
+c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda
+(H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
+(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1
+u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1
+c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead
+c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P
+H5))))))))))))) i v y x H0))) H)))))).
+
+theorem csubst0_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
+(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
+u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))))
+(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
+c2))))))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1)
+x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda
+(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_:
+nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j
+v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_:
+nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j:
+nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y
+x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda
+(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k
+j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda
+(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2:
+T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_:
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda
+(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2:
+T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C
+(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c |
+(CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let
+H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0
+u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match
+e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
+t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq
+K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))
+(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3))))
+(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2:
+C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda
+(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k
+u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
+u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1
+c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1
+u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda
+(_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda
+(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C
+(CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j:
+nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda
+(c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_:
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
+(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3))))
+(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2:
+C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
+(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k
+u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
+u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1
+c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0)
+(s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1
+k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0
+(refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c
+H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0:
+C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0
+c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda
+(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda
+(u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u)
+(CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _)
+\Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u)
+(CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0
+c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda
+(_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda
+(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C
+(CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def
+(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda
+(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
+c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3
+T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
+c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0
+i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda
+(_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda
+(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
+(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2))))
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
+(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
+c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0)
+(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3
+k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0
+(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u
+H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0:
+T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0
+u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0
+c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda
+(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_:
+nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j
+v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda
+(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k
+u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _)
+\Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0)
+(CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0
+c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to
+(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j))))
+(\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3:
+T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
+j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0
+(\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0
+(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1:
+K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k
+j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k
+u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda
+(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k
+u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
+u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
+c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat
+(s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2)
+(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))))
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
+(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1))))
+(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k
+j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k
+u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j:
+nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda
+(_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k
+u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
+u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
+c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12
+H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))).
+
+theorem csubst0_gen_S_bind_2:
+ \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall
+(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to
+(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x
+(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2))
+(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
+C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_:
+T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1
+(Bind b) v1))))))))))))
+\def
+ \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b)
+v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x
+c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda
+(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i
+v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1:
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1:
+T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0:
+(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x
+y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C
+x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
+T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1))))))))
+(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda
+(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i))
+\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1:
+T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1))))
+(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead
+c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1
+v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1:
+C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k:
+K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat
+(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b)
+v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0]))
+(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K
+(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2
+(Bind b) v2) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
+\Rightarrow t])) (CHead c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8:
+(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0:
+C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
+(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i
+v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2))))
+(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
+(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1:
+T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T
+u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K
+k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K
+(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2))
+(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C
+(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0
+u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
+T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1
+(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e
+in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n)
+\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda
+(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1:
+T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead
+c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda
+(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1:
+C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1:
+T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T
+(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind
+b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b)
+u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0:
+nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2:
+(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead
+c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2))
+(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
+C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2))))
+(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
+(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
+T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq
+nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b)
+v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
+(CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K
+(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2
+(Bind b) v2) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9:
+(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t:
+T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
+(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i
+v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2))))
+(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
+(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
+T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C
+c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2))
+\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
+c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
+(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
+C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
+T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead
+c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c:
+C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda
+(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b)
+(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda
+(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
+C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3
+(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1
+v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3:
+C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let
+H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda
+(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i)
+H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to
+((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i
+v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C
+(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3
+(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1
+v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3:
+C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in
+(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i
+H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda
+(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C
+(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind
+b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
+T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3
+c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead
+c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
+(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16
+(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7))
+H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda
+(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1:
+C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq
+nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda
+(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b)
+v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C
+c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
+T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3
+c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b)
+v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C
+(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0
+| (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6)
+in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u2) (CHead c2
+(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0
+c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C
+c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1
+v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
+C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2))))
+(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
+(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
+T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def
+(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14
+\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15
+\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10)
+in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0
+i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b)
+v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C
+(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
+C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
+T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1
+k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda
+(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0
+| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0
+(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to
+(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1
+(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
+(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
+C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
+T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead
+c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda
+(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0
+(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T
+(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind
+b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3
+c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2))))
+(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
+(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
+T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3:
+C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
+T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18
+(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8))
+H7)))))))))))))) y0 v x y H1))) H0))) H))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst0/clear.ma".
+
+include "LambdaDelta-1/csubst0/drop.ma".
+
+include "LambdaDelta-1/getl/fwd.ma".
+
+theorem csubst0_getl_ge:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1
+e) \to (getl n c2 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all
+c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0:
+C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1
+x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5:
+(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4))
+(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0:
+nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0:
+nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e))
+(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9:
+(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat
+x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_:
+(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
+(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3)
+H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9))
+(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x
+(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0)
+x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda
+(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e
+(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e
+(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F
+C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O
+c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O
+c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13:
+(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e))
+H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4)
+H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e
+x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n
+i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))).
+
+theorem csubst0_getl_lt:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1
+e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all
+c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0:
+C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x:
+C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def
+(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K
+C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O
+c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B
+C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e)
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6:
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u
+w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k
+n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0:
+K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x
+(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda
+(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda
+(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop
+n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear
+(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b:
+B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12:
+(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1
+(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4
+(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2
+(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0
+(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead
+e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2
+(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12))
+e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n
+O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v
+x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n
+c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1
+(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3)))))))
+x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind
+K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1
+e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0:
+K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x
+(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda
+(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda
+(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop
+n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear
+(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b:
+B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12:
+(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1
+(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4
+(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2
+(CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0
+(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead
+e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n
+c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3))
+H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11:
+(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s
+(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let
+H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12
+(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1
+x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear
+x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C
+T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4
+B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e
+H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2
+(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n))
+v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4:
+B).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H17: (eq C e
+(CHead x5 (Bind x4) x6))).(\lambda (H18: (clear x2 (CHead x5 (Bind x4)
+x7))).(\lambda (H19: (subst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x5
+(Bind x4) x6) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1
+(getl n c2 (CHead x5 (Bind x4) x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
+x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b)
+u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5
+x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind
+x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18
+f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda
+(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18:
+(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v
+x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2
+c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4)
+x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4)
+x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))
+(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7
+(refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4)
+x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f
+x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda
+(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq
+C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4)
+x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20:
+(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7)
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
+(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7))
+(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11
+(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17))))))))))
+H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
+n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k
+n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n
+c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda
+(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq
+C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda
+(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s
+x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
+(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4))
+\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1
+x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2
+(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3
+x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda
+(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3)
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u))))))
+(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u)))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
+(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead
+e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda
+(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C
+T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4
+(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4)
+(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind
+b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2
+(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3
+x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda
+(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n)
+(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy
+i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e
+(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n))
+v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18:
+(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e
+(CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18:
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+(minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2
+(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e)
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda
+(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5)
+x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21:
+(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7)
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1
+(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b)
+u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6
+x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind
+x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20
+f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda
+(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20:
+(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v
+x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2
+c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5)
+x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5)
+x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))
+(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8
+(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5)
+x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f
+x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda
+(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq
+C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5)
+x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22:
+(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8)
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
+(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
+x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
+x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8))
+(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12
+(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19))))))))))
+H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
+
+theorem csubst0_getl_ge_back:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2
+e) \to (getl n c1 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all
+c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0:
+C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2
+x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5:
+(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3)
+H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0:
+nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0:
+nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e))
+(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O
+c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1
+e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9:
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat
+x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_:
+(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
+(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2)
+H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9))
+(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x
+(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0)
+x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda
+(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e
+(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v
+H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9:
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i
+O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O
+c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13:
+(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e))
+H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3)
+H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0
+x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n
+i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))).
+
+theorem csubst0_getl_lt_back:
+ \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2
+e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1
+e2)) (\lambda (e1: C).(getl n c1 e1))))))))))))
+\def
+ \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def
+(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e))
+(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda
+(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x
+\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in
+(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x))
+(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda
+(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))
+(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0
+(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda
+(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or
+(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2))
+(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0
+(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def
+(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in
+(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2))
+(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda
+(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1:
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))
+(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1:
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind
+C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0
+e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1
+e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11:
+(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl
+n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1:
+C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1
+e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8
+H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst0/defs.ma".
+
+theorem csubst0_snd_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c
+(Bind b) u1) (CHead c (Bind b) u2))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind
+b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b)
+u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S
+i))))))))).
+
+theorem csubst0_fst_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall
+(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1
+(Bind b) u) (CHead c2 (Bind b) u))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind
+b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b)
+u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))).
+
+theorem csubst0_both_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i
+v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b)
+u2))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n:
+nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))
+(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S
+i))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst0/defs.ma".
+
+inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def
+| csubst1_refl: csubst1 i v c1 c1
+| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst1/defs.ma".
+
+include "LambdaDelta-1/csubst0/fwd.ma".
+
+include "LambdaDelta-1/subst1/props.ma".
+
+theorem csubst1_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
+(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T
+C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 i v c1 c2))))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1)
+x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C
+(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2:
+C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_:
+C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1
+c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl
+i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1)
+c2)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i)
+(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2))))
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda
+(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C
+(\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3)))) (\lambda (H1: (ex3_2 T nat (\lambda (_: T).(\lambda
+(j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C
+c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s
+k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2))))
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda
+(u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H2:
+(eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead c1 k x0))).(\lambda
+(H4: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2
+T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda
+(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 (\lambda (n:
+nat).(subst0 n v u1 x0)) H4 i (s_inj k i x1 H2)) in (ex3_2_intro T C (\lambda
+(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda
+(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single
+i v u1 x0 H5) (csubst1_refl i v c1))) c2 H3)))))) H1)) (\lambda (H1: (ex3_2 C
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
+j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3
+k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1:
+nat).(\lambda (H2: (eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead
+x0 k u1))).(\lambda (H4: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1)
+(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead
+c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda
+(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1
+(\lambda (n: nat).(csubst0 n v c1 x0)) H4 i (s_inj k i x1 H2)) in
+(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1)
+(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C
+(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H5))) c2
+H3)))))) H1)) (\lambda (H1: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k
+u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (x2: nat).(\lambda (H2: (eq nat (s k i) (s k x2))).(\lambda (H3:
+(eq C c2 (CHead x1 k x0))).(\lambda (H4: (subst0 x2 v u1 x0)).(\lambda (H5:
+(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C
+(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x2 (\lambda (n:
+nat).(csubst0 n v c1 x1)) H5 i (s_inj k i x2 H2)) in (let H7 \def (eq_ind_r
+nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H4 i (s_inj k i x2 H2)) in
+(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0)
+(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C
+(CHead x1 k x0)) (subst1_single i v u1 x0 H7) (csubst1_sing i v c1 x1 H6))))
+c2 H3)))))))) H1)) (csubst0_gen_head k c1 c2 u1 v (s k i) H0)))) x H))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst1/props.ma".
+
+include "LambdaDelta-1/csubst0/getl.ma".
+
+include "LambdaDelta-1/subst1/props.ma".
+
+include "LambdaDelta-1/drop/props.ma".
+
+theorem csubst1_getl_ge:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1
+e) \to (getl n c2 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to
+(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
+(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
+(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))).
+
+theorem csubst1_getl_lt:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1
+e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2:
+C).(getl n c2 e2)))))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to
+(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl
+n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S
+(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1
+e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1
+(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H))))
+(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda
+(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0:
+nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n
+c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind
+(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u:
+T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)))
+(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S
+(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl
+(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S
+(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1
+(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0)
+x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1
+(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
+n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2:
+C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
+v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus
+i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1
+(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3:
+C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1
+(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1
+(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0)
+x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1
+(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
+n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2:
+C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
+v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus
+i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T
+T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda
+(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda
+(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
+(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2:
+C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind
+x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7:
+(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1
+x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2))))
+(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind
+x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4)
+(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind
+x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1
+H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))).
+
+theorem csubst1_getl_ge_back:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2
+e) \to (getl n c1 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to
+(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
+(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
+(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))).
+
+theorem getl_csubst1:
+ \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c
+(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_:
+C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0
+a))))))))
+\def
+ \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e:
+C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind
+(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind
+Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0
+a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda
+(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n)
+(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2
+C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda
+(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda
+(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to
+(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda
+(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind
+(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O
+(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0:
+C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t:
+T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b)
+t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
+(CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b)
+t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind
+b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B)
+with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e
+(Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind
+Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in
+((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
+(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e
+(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u)
+H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t
+(\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0
+(CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0
+a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda
+(_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda
+(a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_:
+C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a:
+C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead
+c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u
+H3)))) H2)) H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e:
+C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind
+Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T
+(\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0:
+C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2:
+(subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead
+e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr)
+u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in
+(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda
+(a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda
+(a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0
+x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0
+(Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))
+(CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O
+x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3))))
+H1)))))))) k)))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall
+(e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind
+(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e
+(Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S
+n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))
+(\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n)
+(CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind
+Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u
+(CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0
+a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u:
+T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0:
+C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a:
+C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0:
+K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0
+t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_:
+C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a:
+C).(drop (S O) (S n) a0 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
+(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead
+e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in
+(ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
+T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u
+(getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda
+(_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda
+(a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda
+(H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C
+C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t)
+a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0
+(Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift
+(S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4))))
+H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
+T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr)
+u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T
+(\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
+T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e
+u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0:
+C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0
+x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u
+(CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S
+n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x)
+(csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat
+(S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst1/defs.ma".
+
+include "LambdaDelta-1/subst1/defs.ma".
+
+theorem csubst1_head:
+ \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
+v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2))))))))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t:
+T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k
+i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c:
+C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i)
+v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1
+c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i
+c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
+t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1)
+(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2)
+(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v
+c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both
+k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))).
+
+theorem csubst1_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
+v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b)
+u2))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n:
+nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))
+(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S
+i))))))))))).
+
+theorem csubst1_flat:
+ \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
+v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f)
+u2))))))))))
+\def
+ \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n:
+nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2)))
+(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/defs.ma".
+
+include "LambdaDelta-1/clear/fwd.ma".
+
+theorem csubt_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2))
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2))
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
+(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u)
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear
+(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind
+b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2))))
+(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda
+(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g
+c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))))
+(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def
+(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g
+e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1
+e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x:
+C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C
+(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f)
+u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
+C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall
+(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda
+(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3:
+(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1)
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2:
+C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt
+g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b)
+u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2)
+(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3))))))))))))
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_:
+((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2))
+(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda
+(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda
+(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst)
+t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2:
+C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4
+(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2
+H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1
+c2 H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/arity.ma".
+
+theorem csubt_csuba:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba
+g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda
+(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
+(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda
+(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b:
+B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u:
+T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u
+t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A
+(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g
+a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda
+(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g
+x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1)))))
+H4))))))))))) c1 c2 H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/defs.ma".
+
+inductive csubt (g: G): C \to (C \to Prop) \def
+| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n))
+| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall
+(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u))))))
+| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall
+(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g
+(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/fwd.ma".
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+theorem csubt_drop_flat:
+ \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall
+(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
+(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
+\def
+ \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0:
+nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1:
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f)
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1
+(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H
+(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let
+H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C
+(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g
+d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O
+c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x
+(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u)
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f)
+u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda
+(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2)
+\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u))
+\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2
+(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda
+(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall
+(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead
+d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u:
+T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind
+(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1)
+(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort
+n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5
+\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
+in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop
+(S n0) O (CSort n1) (CHead d2 (Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0)
+O (CHead d1 (Flat f) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda
+(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop
+(S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
+u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall
+(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f)
+u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S
+n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda
+(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead
+c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g
+d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g
+d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x
+(Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1
+(Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1:
+C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u)
+(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0)
+u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1
+x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead
+x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1
+(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u:
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
+u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S
+n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f)
+u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O
+(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5:
+(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x
+(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead
+d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u:
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
+u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u
+t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda
+(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f)
+u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0
+O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f)
+u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O
+c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f)
+u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0
+c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0
+H5)))))))))))))) c1 c2 H0)))))) n))).
+
+theorem csubt_drop_abbr:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g
+c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop
+n O c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1:
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u:
+T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr)
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1
+(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in
+(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq
+C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2
+(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr)
+u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u)
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead
+d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3))))
+H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2:
+C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda
+(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda
+(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O
+(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr)
+u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr)
+u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3:
+(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0)
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1)
+(CHead d2 (Bind Abbr) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1
+(Bind Abbr) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S
+n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr)
+u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall
+(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind
+Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda
+(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop
+(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2
+(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda
+(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x
+(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4
+(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1
+u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3))))))))
+(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda
+(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr)
+u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S
+n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr)
+u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S
+n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind
+Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u)))))
+(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0
+H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g
+c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda
+(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0
+(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead
+x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0
+(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda
+(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1:
+C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead
+d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g
+c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind
+Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2
+(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda
+(H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
+(CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind
+Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1
+(Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
+
+theorem csubt_drop_abst:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g
+c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind
+Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1:
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t:
+T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst)
+t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
+g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g
+c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1
+(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H
+(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in
+(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2:
+C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)))
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr)
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_:
+C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3
+g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C
+(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt
+g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t)))
+(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda
+(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead
+d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2:
+C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead
+d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t)))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead
+d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3))
+(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2)))
+(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2:
+T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t))))
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead
+x0 (Bind Abbr) x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g d1
+x1 t)).(\lambda (H7: (ty3 g x0 x1 t)).(eq_ind_r C (CHead x0 (Bind Abbr) x1)
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O
+O c (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind
+Abbr) x1) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda
+(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0
+(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))
+(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
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+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
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+\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
+in (False_ind (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
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+u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda
+(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1
+t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2)
+(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t)))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
+g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7
+u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead
+d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t:
+T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst)
+t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_:
+(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0:
+T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind
+Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g
+d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_:
+C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3
+g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2
+(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))))
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7:
+(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst)
+t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))))
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
+(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind
+Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda
+(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3
+(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0
+t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda
+(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0
+t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S
+n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))))
+(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda
+(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7:
+(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr)
+x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1
+t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))))
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_:
+C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3
+g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1)
+H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0
+(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/defs.ma".
+
+theorem csubt_gen_abbr:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g
+(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr)
+v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2:
+C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))
+(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c:
+C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind
+Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C
+(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1
+| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3)
+in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind
+Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1
+e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k
+t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K
+(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v)
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt
+g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g
+c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr)
+v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3
+(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1
+c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
+e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1
+(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
+\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr)
+v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2)
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)))))))))))
+(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_:
+(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u
+t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr)
+v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C
+(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v)))
+(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))).
+
+theorem csubt_gen_abst:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g
+(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda
+(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda
+(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind
+Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1
+e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_:
+C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3
+g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or
+(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
+C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1:
+(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst)
+v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
+T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))
+H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1
+c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
+(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
+(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1
+(Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
+\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
+(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
+(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2:
+C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g
+e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda
+(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2))))
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
+v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind
+Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
+v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl
+(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst)
+v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
+(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
+T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
+(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3
+(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1
+(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
+v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1
+(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
+\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst)
+v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b)
+u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T
+(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
+(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3:
+C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind
+Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
+v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u
+t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst)
+v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind
+Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def
+(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def
+(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
+C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind
+C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda
+(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda
+(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
+(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind
+Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt
+g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind
+Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))).
+
+theorem csubt_gen_flat:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall
+(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C
+c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead
+e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C
+(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
+e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda
+(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f)
+v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
+e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k
+u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _)
+\Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u)
+(CHead e1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v
+(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat
+f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda
+(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v)))
+(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in
+(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
+(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f)
+v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f)
+v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3:
+C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f)
+v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda
+(e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind
+Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind
+Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2:
+C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda
+(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
+e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u
+t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t)
+(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2:
+C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))).
+
+theorem csubt_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
+(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
+(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b:
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1)
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1
+(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
+H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9)
+in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind
+b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b)
+u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2))
+H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3
+B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
+e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u
+t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst)
+t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _
+_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
+((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
+((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead
+c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst
+b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0:
+T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0:
+T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c:
+C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
+H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10)
+in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b)
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
+C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
+(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
+e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15))))))))))
+H7)) H6))))))))))) y c2 H0))) H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/clear.ma".
+
+include "LambdaDelta-1/csubt/drop.ma".
+
+include "LambdaDelta-1/getl/clear.ma".
+
+theorem csubt_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
+(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g
+c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n
+c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e:
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u)))
+(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x:
+C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
+Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1
+(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
+u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda
+(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1
+(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
+u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
+Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1
+(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr)
+u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear
+(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1
+c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2
+(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1
+(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1
+(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
+\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
+Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u)
+t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda
+(c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda
+(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
+(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr
+H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c
+(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2
+(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n
+c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1
+u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7)))))
+(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda
+(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5
+in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to
+(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda
+(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall
+(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1:
+C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
+C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c:
+C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat
+f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u)
+(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def
+(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u)
+H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2))
+(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2:
+C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13:
+(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C
+(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt
+g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2
+(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def
+(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15)
+in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u)
+c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda
+(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall
+(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1:
+C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
+C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0
+(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_:
+B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t)))))
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
+(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4:
+T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0
+O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10
+(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead
+x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4)
+x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5
+x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7
+(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5
+(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21
+\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl
+n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22
+(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23)))))
+H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1
+H2)))) H0))))))).
+
+theorem csubt_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall
+(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g
+c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e:
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t)))
+(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))
+(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead
+d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c
+(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1
+(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst)
+t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0:
+C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t))
+\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1
+(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst)
+t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear
+(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1
+c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n
+c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5:
+(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b)
+t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
+(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b)
+t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0)
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind
+Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t)
+t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1
+x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r
+T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let
+H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t)))
+H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1
+(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2
+(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O c2
+(CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))
+(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2
+(CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t)
+(CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda
+(H16: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
+(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O
+c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20:
+(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2
+H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18
+(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12
+d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1
+(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead
+d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n
+O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1:
+C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1
+c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl
+n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda
+(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9:
+(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2))
+H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in
+(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0
+(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g
+(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C
+(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2:
+C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2:
+C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13:
+(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C
+(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt
+g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2)))
+(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t)))
+(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2
+(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
+Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
+(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3
+(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2
+(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead
+d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O
+c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15:
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr)
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_:
+C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3
+g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind
+Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4
+t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c:
+C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
+Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
+(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_:
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3
+(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14)))))
+H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1
+(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2:
+C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9:
+(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10:
+(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9)
+in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1
+(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
+T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2
+C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12:
+(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0
+(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind
+x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4)
+e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda
+(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4)
+x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5
+x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda
+(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5
+(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def
+(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18)
+in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
+(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead
+d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t)))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2
+(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt
+g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2
+C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24:
+(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2:
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst)
+t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
+g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
+C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2
+x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C
+T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
+(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
+(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g
+d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9:
+C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0
+x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda
+(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2))
+(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda
+(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))
+(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
+g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr)
+x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14)))))))
+H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/getl.ma".
+
+include "LambdaDelta-1/pc3/left.ma".
+
+theorem csubt_pr2:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2:
+C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
+C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0
+c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_:
+(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta
+c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
+
+theorem csubt_pc3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t:
+T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4:
+T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1
+c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1
+c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4
+(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1
+t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall
+(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda
+(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0
+t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/defs.ma".
+
+theorem csubt_refl:
+ \forall (g: G).(\forall (c: C).(csubt g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0))
+(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0
+c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/pc3.ma".
+
+include "LambdaDelta-1/csubt/props.ma".
+
+theorem csubt_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda
+(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda
+(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u:
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2:
+C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3
+t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1
+c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda
+(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c
+c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda
+(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g
+d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c
+c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda
+(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
+u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5:
+(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g
+n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
+Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2:
+((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2:
+C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0
+c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2:
+C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0
+u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n)
+(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2))
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda
+(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst)
+u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6:
+(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g
+n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0
+u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T
+(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda
+(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
+T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3
+g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr)
+x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g
+n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2:
+C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda
+(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0
+t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1
+c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b)
+u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_:
+(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2
+w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind
+Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v
+(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda
+(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4:
+T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c
+c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c
+c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
+theorem csubt_ty3_ld:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u
+v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1
+t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H:
+(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead
+c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead
+c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubv/defs.ma".
+
+include "LambdaDelta-1/clear/fwd.ma".
+
+theorem csubv_clear_conf:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1:
+B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to
+(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
+d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2
+(Bind b2) v2))))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall
+(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2:
+B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2)
+v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda
+(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1)
+v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2:
+B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2)
+v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3
+c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear
+c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2:
+C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
+C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0:
+T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1)
+v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c]))
+(CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3
+(CHead d1 (Bind b1) v0) v1 H2)) in ((let H4 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b1 |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
+b) \Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0)
+(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1
+H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow
+t])) (CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void
+c3 (CHead d1 (Bind b1) v0) v1 H2)) in (\lambda (_: (eq B b1 Void)).(\lambda
+(H7: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda (b2: B).(\lambda
+(d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind b2)
+v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_:
+T).(csubv c3 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear
+(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))) Void c4 v2 H0
+(clear_bind Void c4 v2)) d1 H7)))) H4)) H3)))))))))))) (\lambda (c3:
+C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (_: ((\forall (b1:
+B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind b1) v1)) \to
+(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
+d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2
+(Bind b2) v2)))))))))))).(\lambda (b1: B).(\lambda (_: (not (eq B b1
+Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b0:
+B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear (CHead c3 (Bind b1)
+v1) (CHead d1 (Bind b0) v0))).(let H4 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
+(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1)
+v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in ((let H5 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+b0])])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3
+(CHead d1 (Bind b0) v0) v1 H3)) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 |
+(CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1)
+v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in (\lambda (_: (eq
+B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B
+C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda
+(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2)
+(CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda
+(d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda (b3: B).(\lambda (d2:
+C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind b3)
+v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) H4)))))))))))))))
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (H1:
+((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear
+c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda (f1: F).(\lambda (f2:
+F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1:
+C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1
+(Bind b1) v0))).(let H_x \def (H1 b1 d1 v0 (clear_gen_flat f1 c3 (CHead d1
+(Bind b1) v0) v1 H2)) in (let H3 \def H_x in (ex2_3_ind B C T (\lambda (_:
+B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2:
+B).(\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 (Bind b2) v3)))))
+(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
+d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4
+(Flat f2) v2) (CHead d2 (Bind b2) v3)))))) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H4: (csubv d1 x1)).(\lambda (H5: (clear c4
+(CHead x1 (Bind x0) x2))).(ex2_3_intro B C T (\lambda (_: B).(\lambda (d2:
+C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
+C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2)
+v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind x0) x2) H5 f2 v2)))))))
+H3))))))))))))))) c1 c2 H))).
+
+theorem csubv_clear_conf_void:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1:
+C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda
+(v2: T).(clear c2 (CHead d2 (Bind Void) v2))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c
+(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2
+(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1:
+T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort
+(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead
+d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0:
+(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3
+(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2
+(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1:
+C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1
+(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
+\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind Void) v1)
+(clear_gen_bind Void c3 (CHead d1 (Bind Void) v0) v1 H2)) in ((let H4 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind
+Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind
+Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c:
+C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2:
+C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void)
+v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2)))
+(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2
+(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3)))))))))))
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_:
+((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to
+(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2:
+C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda
+(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1:
+T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear
+(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Void) v0)
+(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1
+H3)) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Void | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead d1 (Bind Void) v0)
+(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1
+H3)) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow
+t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3
+(CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq B Void b1)).(\lambda
+(H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda (v3: T).(clear
+(CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 \def (eq_ind_r
+B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let H10 \def (match
+(H9 (refl_equal B Void)) in False return (\lambda (_: False).(ex2_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) (\lambda (d2: C).(\lambda
+(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) with
+[]) in H10)) d1 H8)))) H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1:
+T).((clear c3 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear
+c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda (f1: F).(\lambda (f2:
+F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0:
+T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind Void)
+v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind Void) v0)
+v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2
+(Bind Void) v3)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1
+d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead
+d2 (Bind Void) v3))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (csubv
+d1 x0)).(\lambda (H5: (clear c4 (CHead x0 (Bind Void) x1))).(ex2_2_intro C T
+(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda
+(v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4
+(clear_flat c4 (CHead x0 (Bind Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1
+c2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+inductive csubv: C \to (C \to Prop) \def
+| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n))
+| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall
+(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind
+Void) v2))))))
+| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall
+(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall
+(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2)))))))))
+| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall
+(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1
+(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubv/props.ma".
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+theorem csubv_drop_conf:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1:
+C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1
+e2)) (\lambda (e2: C).(drop h O c2 e2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h
+O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O
+c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda
+(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq
+nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O
+(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C
+(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2))))
+(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2))
+(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n)
+(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort
+n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3
+c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to
+(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4
+e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h:
+nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda
+(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2:
+C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2)
+e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C
+(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c
+e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C
+(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop
+O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same
+c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl
+(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0
+O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2))
+(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3:
+(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind
+Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in
+(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4
+e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O
+(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1
+x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1
+e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5
+(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall
+(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2:
+C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1:
+B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1:
+T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h
+O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead
+c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2:
+C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead
+c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2
+C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind
+b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1)
+e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind
+b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2)
+v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0:
+nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C
+(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind
+b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1)
+e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1
+h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2))
+(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2))
+(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x:
+C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C
+(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4
+(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h
+H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3
+c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to
+(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4
+e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat
+f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1)
+e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O
+(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1)
+v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2)
+e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2))
+(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2)
+v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1
+(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda
+(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2:
+C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2)
+e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x
+\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in
+(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2:
+C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda
+(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda
+(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda
+(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2)
+v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h
+H2)))))))))))) c1 c2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubv/clear.ma".
+
+include "LambdaDelta-1/csubv/drop.ma".
+
+include "LambdaDelta-1/getl/fwd.ma".
+
+theorem csubv_getl_conf:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1:
+B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl
+i c2 (CHead d2 (Bind b2) v2)))))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1:
+B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i
+c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind
+b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e:
+C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda
+(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
+C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x:
+C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1)
+v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in
+(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2))
+(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
+d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead
+d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda
+(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3)
+in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2:
+C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
+C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda
+(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2)
+v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8:
+(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro
+B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))))
+(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind
+b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6
+H9))))))) H7)))))) H4)))))) H1))))))))).
+
+theorem csubv_getl_conf_void:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1:
+C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1))
+\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2:
+C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1:
+C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1
+(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i
+H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e
+(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2
+(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda
+(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1
+c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2))
+(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2
+(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda
+(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1
+H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_:
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2
+(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1
+d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void)
+v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1
+x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T
+(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda
+(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2
+(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubv/defs.ma".
+
+include "LambdaDelta-1/T/props.ma".
+
+theorem csubv_bind_same:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall
+(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b)
+v2)))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b:
+B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1
+(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2:
+T).(csubv_bind c1 c2 H Abbr (\lambda (H0: (eq B Abbr Void)).(not_abbr_void
+H0)) Abbr v1 v2))) (\lambda (v1: T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst
+(sym_not_eq B Void Abst not_void_abst) Abst v1 v2))) (\lambda (v1:
+T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))).
+
+theorem csubv_refl:
+ \forall (c: C).(csubv c c)
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n:
+nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda
+(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead
+c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t)))
+(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/tlt/defs.ma".
+
+include "LambdaDelta-1/iso/defs.ma".
+
+include "LambdaDelta-1/clen/defs.ma".
+
+include "LambdaDelta-1/flt/defs.ma".
+
+include "LambdaDelta-1/app/defs.ma".
+
+include "LambdaDelta-1/cnt/defs.ma".
+
+include "LambdaDelta-1/cimp/defs.ma".
+
+include "LambdaDelta-1/csubv/defs.ma".
+
+include "LambdaDelta-1/subst/defs.ma".
+
+include "LambdaDelta-1/subst1/defs.ma".
+
+include "LambdaDelta-1/csubst1/defs.ma".
+
+include "LambdaDelta-1/fsubst0/defs.ma".
+
+include "LambdaDelta-1/next_plus/defs.ma".
+
+include "LambdaDelta-1/sty1/defs.ma".
+
+include "LambdaDelta-1/llt/defs.ma".
+
+include "LambdaDelta-1/aprem/defs.ma".
+
+include "LambdaDelta-1/ex0/defs.ma".
+
+include "LambdaDelta-1/wcpr0/defs.ma".
+
+include "LambdaDelta-1/csuba/defs.ma".
+
+include "LambdaDelta-1/nf2/defs.ma".
+
+include "LambdaDelta-1/ex2/defs.ma".
+
+include "LambdaDelta-1/csubc/defs.ma".
+
+include "LambdaDelta-1/pc1/defs.ma".
+
+include "LambdaDelta-1/ex1/defs.ma".
+
+include "LambdaDelta-1/csubt/defs.ma".
+
+include "LambdaDelta-1/wf3/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+include "LambdaDelta-1/lift/defs.ma".
+
+include "LambdaDelta-1/r/defs.ma".
+
+inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
+| drop_refl: \forall (c: C).(drop O O c c)
+| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e:
+C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u)
+e))))))
+| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c:
+C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d)
+(CHead c k (lift h (r k d) u)) (CHead e k u)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/drop/defs.ma".
+
+theorem drop_gen_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop
+h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x:
+C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c:
+C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d
+O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n))
+\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C
+c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq
+nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal
+nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda
+(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n))
+\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C
+(CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
+(CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O)
+(eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0:
+nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c
+e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq
+nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0
+(r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0)
+u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort
+_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
+(False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq
+nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))).
+
+theorem drop_gen_refl:
+ \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
+\def
+ \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O
+(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y:
+nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y
+n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0:
+nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0:
+nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to
+(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq
+nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c:
+C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq
+nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda
+(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S
+h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C
+(CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c
+e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c
+e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat
+h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in
+(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat
+n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h
+(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d)
+(\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let
+H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4)
+in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h
+H6)))))))))))))) y y0 x e H1))) H0))) H))).
+
+theorem drop_gen_drop:
+ \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
+nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
+nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u)
+(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x))
+(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda
+(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to
+(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y
+x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_:
+nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x))))
+(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h))
+\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c
+c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq
+nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u)
+(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda
+(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
+True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c
+(CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda
+(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4:
+(((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to
+(drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S
+h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c
+k u))).(let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
+\Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let H9 \def
+(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
+(CHead c k u) H7) in ((let H10 \def (f_equal C T (\lambda (e0: C).(match e0
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H7) in (\lambda (H11: (eq K
+k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1:
+C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u))
+\to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda
+(c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0
+(\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c
+(CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def
+(eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17
+\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_:
+nat).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in
+(let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to
+((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h
+H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e))
+H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda
+(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3:
+(drop h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0
+d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c
+e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq
+nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead
+c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0
+(lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind
+nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq
+C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let
+H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h)
+H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
+\Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in
+((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
+(CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \def
+(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t:
+T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0
+u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S
+h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h)
+(r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15:
+(eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S
+h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h)
+(r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1:
+C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0
+(\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19
+\def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1
+d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k
+H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c
+e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead
+e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h))
+\to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d)
+c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \def (eq_ind nat (S d)
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r
+k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11))))))))))))))))
+y1 y0 y x H2))) H1))) H0))) H)))))).
+
+theorem drop_gen_skip_r:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda
+(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k
+d) e c)))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k
+u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0))
+(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d)
+u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0:
+(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y))
+(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x
+(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c)))))
+(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d))
+\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k
+(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda
+(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k
+u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C
+c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))))
+(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e
+k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3))))
+(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda
+(H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e
+(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0
+h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
+c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C
+e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d))
+\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
+(lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
+c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1:
+C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat
+O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2
+C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d)
+u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda
+(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
+C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
+(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0
+(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
+c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5:
+(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
+(CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let
+H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
+(CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda
+(H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t:
+T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead
+e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let
+H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1
+(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k
+d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12
+\def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in
+(let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
+((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
+(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9)
+in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12
+k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead
+c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0:
+C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0:
+nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow d0 |
+(S n) \Rightarrow n])) (S d0) (S d) H4) in (let H16 \def (eq_ind nat d0
+(\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to
+(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda
+(e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat
+d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d
+(\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n)
+u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
+c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u))
+(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c))
+c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9)))))
+u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))).
+
+theorem drop_gen_skip_l:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T
+(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop h (r k d) c e))))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u)
+x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda
+(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0:
+(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x))
+(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e:
+C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
+d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq
+nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e:
+C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k
+d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda
+(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1:
+C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind
+(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (CHead e k
+v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda
+(e: C).(\lambda (_: T).(drop O (r k d) c e)))) H4)) c0 H3)))) (\lambda (k0:
+K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r
+k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u))
+\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c
+e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq
+C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
+(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let
+H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
+(CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow
+u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H5) in
+(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
+C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to
+(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c
+H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e))
+H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d))
+\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r
+k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d)
+c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r
+k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
+| (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda
+(e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda
+(v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop (S h0) (r k d) c e0)))) H15))))))))) H7)) H6))))))))))) (\lambda
+(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
+C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
+(S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda
+(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u
+(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
+e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda
+(H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0
+(lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0
+d0) u0)) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let
+rec lref_map (f: ((nat \to nat))) (d1: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0)
+\Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in
+lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in
+(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
+C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u))
+\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def
+(eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13
+\def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9)
+in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
+((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C
+e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d)
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9)
+in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k
+H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda
+(v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
+(r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k
+d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0:
+C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
+(r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k
+d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C
+(CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
+e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat
+return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n]))
+(S d0) (S d) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat
+(r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T
+(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda
+(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19
+\def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in
+(eq_ind_r nat d (\lambda (n: nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v:
+T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq
+T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h0 (r k d) c e0))))) (ex3_2_intro C T (\lambda (e0: C).(\lambda (v:
+T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq
+T (lift h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T
+(lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7))
+H6)))))))))))) h y0 y x H1))) H0))) H))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+include "LambdaDelta-1/r/props.ma".
+
+theorem drop_skip_bind:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
+d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b)
+(lift h d u)) (CHead e (Bind b) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b)
+d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e
+(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
+
+theorem drop_skip_flat:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
+(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat
+f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat
+f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead
+e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S
+d))))))))).
+
+theorem drop_S:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
+nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
+C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
+T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b)
+u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O
+O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort
+n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O
+(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead
+e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
+(CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1))))
+(drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda
+(H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e
+(Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead
+c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b)
+u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
+\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead
+c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u)
+(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def
+(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t)
+(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0))
+in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0
+(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b)
+(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0
+c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_:
+(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0
+k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b)
+u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0:
+nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b)
+u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+
+theorem drop_ctail:
+ \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop
+h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1)
+(CTail k u c2))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u:
+T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n)
+c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h
+O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0:
+(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n))
+(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u
+(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O
+(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2
+H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2:
+C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k
+u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n
+(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u
+(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop
+n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O
+(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t)
+(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
+(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3
+H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to
+(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail
+k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0:
+K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O
+(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n:
+nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to
+(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail
+k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t)
+c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda
+(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e)))
+(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2:
+(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k
+t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2
+k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k
+x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u
+c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n
+(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop
+h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4
+(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0:
+T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1))))
+(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u)
+x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d)))))))
+c1).
+
+theorem drop_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h
+d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0
+x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2:
+C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq
+nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort
+n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C
+x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1
+(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r
+C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h
+(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d
+(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda
+(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1))))
+(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda
+(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1
+x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t)
+x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2))))))
+(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1)
+\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2)))))
+(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1:
+(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C
+x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t)))
+(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2
+(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_:
+(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t)
+x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t)
+x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O
+(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n
+H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
+(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq
+C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t)
+x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t)
+x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0:
+C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4:
+(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0
+x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4:
+C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7:
+(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r
+C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1
+(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to
+(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0
+(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1
+(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0:
+nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop
+h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k
+n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r
+k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda
+(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0))
+(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n)
+t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n)
+H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k
+x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h
+H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n)
+H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1)))))))
+(drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c).
+
+theorem drop_conf_lt:
+ \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall
+(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h (r k d) c0 e0)))))))))))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u:
+T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
+(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u:
+T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k
+u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop
+h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S
+(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in
+(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda
+(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d))
+x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k
+x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
+h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T
+(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda
+(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda
+(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k
+d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0
+e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall
+(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d))
+c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda
+(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0)
+O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S
+i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind
+(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort
+n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d))
+O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee
+in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda
+(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop
+(S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
+c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1))))))))
+(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0)
+d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k
+d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda
+(k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1
+t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
+(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
+d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
+(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b)
+(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1
+x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b)
+c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T
+u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O
+(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7:
+(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k
+x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0
+x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h
+(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda
+(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
+d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
+(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f)
+(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1
+x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S
+i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
+c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1)
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d)
+x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h
+(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead
+x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r
+k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1)
+H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e
+H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2)))))))))
+k0)))) c)))))) i)).
+
+theorem drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
+(plus d h) i) \to (drop (minus i h) O e a)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c:
+C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e
+a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
+d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda
+(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def
+(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus
+O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5
+\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def
+(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O
+(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0:
+C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h
+H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0:
+nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
+(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a:
+C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d
+h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n:
+nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2:
+(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d
+O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda
+(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n))
+(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6:
+(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O
+O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2
+O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S
+i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0)
+O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0
+a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n)
+c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n))
+H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0)))))
+(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S
+i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e
+a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S
+i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop
+(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1:
+(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t)
+e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h
+n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S
+i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda
+(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0
+(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e
+a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le
+(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop
+(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b)
+c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda
+(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O
+h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0)
+O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a
+c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e
+t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0:
+nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h)
+(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
+(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
+i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind
+b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0)
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop
+(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
+e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0)
+x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0
+(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S
+(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a))
+(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t
+i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h)
+(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e
+H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3)))))))))
+(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat
+f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
+(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S
+i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le
+(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h
+O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind
+(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S
+i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0
+(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0
+(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat
+f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead
+c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead
+c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O
+e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7:
+(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S
+h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0:
+nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h)
+(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
+(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
+i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat
+f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0)
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop
+(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
+e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0)
+x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0
+(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def
+(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0
+(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h))
+(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in
+(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f)
+x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h)
+(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e
+H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3)))))))))
+k)))) c))))) i).
+
+theorem drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
+(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1:
+C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
+\def
+ \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2:
+C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2)
+\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1
+e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1
+e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let
+H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl
+e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1:
+C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda
+(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2:
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O
+c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind
+(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2:
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n:
+nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda
+(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2
+(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1:
+(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O
+O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n)
+H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2))
+(\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S
+j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop
+(S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to
+(ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S
+j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda
+(H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i:
+nat).(\lambda (H0: (drop i O c2 e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0)
+O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
+C).(drop i (S j0) c1 (CHead e2 k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop
+(r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2
+\def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1:
+C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda
+(c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda
+(H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda
+(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2
+(Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind
+b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4
+t))))) H2))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2
+e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C
+(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1
+e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i
+(S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0)
+O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1:
+C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat
+f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0
+x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t)))))
+H2))))) k (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j).
+
+theorem drop_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h
+(minus d i) e1 e2)))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to
+(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to
+(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1
+e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d:
+nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
+nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O
+c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2
+(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda
+(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C
+(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1
+(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0:
+nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1:
+C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
+C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda
+(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind
+(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2:
+C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
+C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0)
+O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h
+O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat
+(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C
+(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S
+i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_:
+(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow
+False])) I (S x) H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1
+e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H4))))) (le_gen_S i0
+O H))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall
+(c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall
+(e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1
+e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda
+(H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0)
+O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n)
+c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2
+(CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S
+i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))
+(\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
+(eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c
+e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq
+nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort
+n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C
+(CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n)
+e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def
+(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in
+(False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
+(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6))))
+(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0))))))))
+(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h
+(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda
+(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0))
+e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
+T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to
+(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S
+i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
+e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h:
+nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
+C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e:
+C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda
+(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O
+(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
+e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0
+(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda
+(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c:
+C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h
+(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O
+(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
+e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop
+h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2
+(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S
+d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda
+(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0)
+O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop
+h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r
+(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2
+(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3
+t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t)
+c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T
+(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1:
+C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus
+(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C
+c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0)
+x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3
+(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in
+(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1:
+C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus
+(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1))
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1:
+C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1))
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x:
+C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S
+i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
+(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S
+i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1))
+H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3)))))))
+(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
+
+theorem drop_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2
+e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
+C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
+nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
+c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h)
+O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d
+(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2
+H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall
+(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall
+(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1
+e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2:
+C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c
+e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
+nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0:
+(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort
+n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda
+(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n)
+e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4)
+in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort
+n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop
+(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda
+(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n)
+(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind
+nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop)
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind
+(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n
+(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2:
+C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d
+(S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda
+(t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall
+(h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O
+c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t)
+e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k
+t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to
+(drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead
+c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda
+(_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O
+c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat
+i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2
+(drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda
+(n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2:
+C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O
+(CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t)
+c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le
+O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O
+(CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k
+(plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r
+k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n)
+(drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0
+n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm
+i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop
+h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le
+d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h:
+nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2:
+C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S
+i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k
+v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda
+(e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O
+(CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3
+(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4:
+(drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall
+(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O
+c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t)
+e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c:
+C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t
+(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k
+x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S
+i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k
+d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S
+(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind
+(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to
+(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r
+(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0
+(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f:
+F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat
+f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4
+(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3)))))))))
+(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/drop/defs.ma".
+
+include "LambdaDelta-1/lift1/defs.ma".
+
+inductive drop1: PList \to (C \to (C \to Prop)) \def
+| drop1_nil: \forall (c: C).(drop1 PNil c c)
+| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds
+c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))).
+
+definition ptrans:
+ PList \to (nat \to PList)
+\def
+ let rec ptrans (hds: PList) on hds: (nat \to PList) \def (\lambda (i:
+nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow
+(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d)
+with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow
+q])))])) in ptrans.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/drop1/defs.ma".
+
+theorem drop1_gen_pnil:
+ \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq
+PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1
+c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda
+(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c
+c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c)))
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds:
+PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to
+(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def
+(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee in PList return
+(\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _)
+\Rightarrow True])) I PNil H4) in (False_ind (eq C c3 c5) H5)))))))))))) y c1
+c2 H0))) H))).
+
+theorem drop1_gen_pcons:
+ \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h:
+nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda
+(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))
+\def
+ \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq
+PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_:
+PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds
+c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind
+(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d
+hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1
+hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d
+hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee in
+PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _
+_ _) \Rightarrow False])) I (PCons h d hds) H1) in (False_ind (ex2 C (\lambda
+(c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 hds c2 c))) H2)))) (\lambda
+(c2: C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1:
+(drop h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2:
+(drop1 hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2
+C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6
+c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let H5
+\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
+(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
+(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat
+(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
+[PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0)
+(PCons h d hds) H4) in ((let H7 \def (f_equal PList PList (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
+\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h
+d hds) H4) in (\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let
+H10 \def (eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds))
+\to (ex2 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6
+c5))))) H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p:
+PList).(drop1 p c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda
+(n: nat).(drop h0 n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda
+(n: nat).(drop n d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h
+d c2 c6)) (\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6))
+H5)))))))))))) y c1 c3 H0))) H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/drop1/fwd.ma".
+
+include "LambdaDelta-1/getl/drop.ma".
+
+theorem drop1_getl_trans:
+ \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
+i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i)
+e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1
+(ptrans hds i) v)))))))))))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
+C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
+C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
+(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl
+(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v))))))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda
+(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in
+(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1))
+(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda
+(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b)
+v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2:
+C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v:
+T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda
+(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i)
+c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda
+(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl
+i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d
+H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3))
+(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match
+(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1))
+(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
+(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
+hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4:
+(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0:
+bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons
+h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
+hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
+(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v))))))
+(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0
+i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow
+(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow
+(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d
+(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)])
+v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def
+(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2:
+C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x
+(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2:
+C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1))
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0:
+C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans
+hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def
+(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b
+x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda
+(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans
+hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S
+(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S
+(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0
+i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans
+hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2
+(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0
+i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1
+x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0
+i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead
+e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i))
+v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0
+i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d)
+false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2:
+C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))
+(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl
+(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))))
+(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8:
+(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let
+H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b)
+(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1
+(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2
+(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans
+hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/drop1/fwd.ma".
+
+include "LambdaDelta-1/drop/props.ma".
+
+include "LambdaDelta-1/getl/defs.ma".
+
+theorem drop1_skip_bind:
+ \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
+C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b)
+(lift1 hds u)) (CHead e (Bind b) u)))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p:
+PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p)
+(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def
+(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead
+c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c
+H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead
+c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda
+(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def
+(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda
+(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S
+n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))
+(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x
+e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b)
+(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e
+(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))).
+
+theorem drop1_cons_tail:
+ \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop
+h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to
+(drop1 (PConsTail hds h d) c1 c3))))))))
+\def
+ \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda
+(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
+c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def
+(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d
+PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0:
+((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
+c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x
+\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C
+(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1
+(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n
+n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3
+(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))).
+
+theorem drop1_trans:
+ \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0)
+\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1
+(papp is1 is2) c1 c2)))))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
+C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2:
+C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1:
+C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2:
+PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def
+(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c:
+C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0:
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0:
+C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0
+c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0:
+C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda
+(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0
+p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1
+c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1
+c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x
+c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1)))))
+H2))))))))))))) is1).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/A/defs.ma".
+
+include "LambdaDelta-1/G/defs.ma".
+
+definition gz:
+ G
+\def
+ mk_G S lt_n_Sn.
+
+inductive leqz: A \to (A \to Prop) \def
+| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
+(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort
+h2 n2))))))
+| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3:
+A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ex0/defs.ma".
+
+include "LambdaDelta-1/leq/defs.ma".
+
+include "LambdaDelta-1/aplus/props.ma".
+
+theorem aplus_gz_le:
+ \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A
+(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n))))))
+\def
+ \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0:
+nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h)
+n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y
+\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n)
+(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0:
+nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A
+(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h:
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A
+(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O
+\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n:
+nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n))
+k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O
+(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0
+n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n)))
+(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0:
+nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S
+(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0))
+(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz
+(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n:
+nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz
+(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S
+k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda
+(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz
+(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0)
+k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a:
+A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0)
+k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S
+n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k).
+
+theorem aplus_gz_ge:
+ \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A
+(aplus gz (ASort h n) k) (ASort (minus h k) n)))))
+\def
+ \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h:
+nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0)
+n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0:
+nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O)
+(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0
+h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda
+(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz
+(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le
+(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0:
+nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n))
+(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0
+x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O
+n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0)
+n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0))
+n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in
+(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus
+gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n))
+k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus
+gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc
+gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)).
+
+theorem next_plus_gz:
+ \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat
+(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0:
+nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat
+S (next_plus gz n n0) (plus n0 n) H))) h)).
+
+theorem leqz_leq:
+ \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz
+(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda
+(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k
+h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2
+(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def
+(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort
+h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1
+(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k)
+(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2)
+(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat)
+on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O |
+(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow
+(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2
+k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in
+(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n:
+nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n:
+nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal
+nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1))
+(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2
+k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a
+(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1
+(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort
+h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus
+(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat
+(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2)
+n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind
+A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return
+(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])
+| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in
+(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1
+k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k
+h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A
+a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1))
+(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2)
+k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort
+(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in
+(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k)
+n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A
+(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k)))
+(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k)))
+n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort
+n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow
+False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1
+n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A
+(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2)
+k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4
+\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O
+(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le
+k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
+nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
+plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus
+(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in
+(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3:
+A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4:
+A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4
+a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
+
+theorem leq_leqz:
+ \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind
+(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus
+h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A
+(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1)))
+(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A
+(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2)))
+(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus
+(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus
+h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2))
+(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n:
+nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus
+h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq
+A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz
+n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1
+h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n:
+nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus
+h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal
+nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2
+n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2)
+(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2)
+(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2)
+(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus
+(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2))
+(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1
+h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0:
+A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0
+a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda
+(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+definition ex1_c:
+ C
+\def
+ CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O).
+
+definition ex1_t:
+ T
+\def
+ THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ex1/defs.ma".
+
+include "LambdaDelta-1/ty3/fwd.ma".
+
+include "LambdaDelta-1/pc3/fwd.ma".
+
+include "LambdaDelta-1/nf2/pr3.ma".
+
+include "LambdaDelta-1/nf2/props.ma".
+
+include "LambdaDelta-1/arity/defs.ma".
+
+include "LambdaDelta-1/leq/props.ma".
+
+theorem ex1__leq_sort_SS:
+ \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc
+g (asucc g (ASort (S (S k)) n))))))
+\def
+ \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g
+(asucc g (ASort (S (S k)) n)))))).
+
+theorem ex1_arity:
+ \forall (g: G).(arity g ex1_c ex1_t (ASort O O))
+\def
+ \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S
+(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O))
+(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)
+O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g
+(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O))
+(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O))
+O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead
+(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst)
+(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort
+O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort
+O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort
+O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O))
+O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))).
+
+theorem ex1_ty3:
+ \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort
+O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t:
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind
+Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst)
+u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
+(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1))
+u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef
+(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda
+(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O
+t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O
+t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O
+x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind
+Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S
+O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_:
+T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort
+O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0
+x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
+x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef
+(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O)))
+O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9:
+T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9)
+x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7
+(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
+x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7
+(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S
+O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x:
+C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr)
+x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
+with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
+[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) |
+(Flat _) \Rightarrow False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
+x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2
+(Bind Abbr) x3) H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda
+(H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0)
+x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
+O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
+x7 (Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def
+(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
+Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
+C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P
+(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7
+(Bind Abst) x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H17)))))
+H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O))
+H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
+(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (\lambda (H3: (ex3_3 C T
+T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: (pc3 (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
+(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex3_2_ind T T
+(\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind
+Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind
+Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S
+O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C
+T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P
+(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
+O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
+x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def
+(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
+Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
+C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P
+(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7
+(Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _)
+\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
+(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C
+T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3)
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O)
+(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda
+(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
+(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def
+(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def
+(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead
+(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst)
+(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort
+O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CSort O)
+(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr)
+x8) (TSort O) H23)) in (False_ind P H24)))))))) H17))))) H14)))))))) H10))
+(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0)
+x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
+O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
+x7 (Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def
+(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
+Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
+C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P
+(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7
+(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _)
+\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
+(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C
+T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3)
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O)
+(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda
+(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
+(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def
+(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def
+(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead
+(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst)
+(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort
+O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x7 | (CHead c _
+_) \Rightarrow c])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst)
+(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O)
+H23)) in ((let H25 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow x8 | (CHead _ _ t) \Rightarrow
+t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O))
+(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in
+(\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind T x8 (\lambda (t:
+T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def (eq_ind T x8 (\lambda
+(t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)) H11 (TSort O)
+H25) in (let H29 \def (eq_ind C x7 (\lambda (c: C).(ty3 g c (TSort O) x9))
+H27 (CSort O) H26) in (or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H30:
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O
+t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind
+Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift
+(S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
+e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g
+e u0 t)))) P (\lambda (x10: C).(\lambda (x11: T).(\lambda (x12: T).(\lambda
+(_: (pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
+O)) (lift (S O) O x12) x4)).(\lambda (H32: (getl O (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr)
+x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def (eq_ind C (CHead x10
+(Bind Abbr) x11) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst
+(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O)
+(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
+O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30))
+(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11:
+T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32:
+(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow x10 | (CHead c _ _) \Rightarrow c])) (CHead x10
+(Bind Abst) x11) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O))
+(CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11)
+H32))) in ((let H35 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow x11 | (CHead _ _ t)
+\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O)
+(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
+x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O)
+(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g
+x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t:
+T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10
+(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst)
+(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
+x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind
+b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S
+O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O)
+O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
+t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O
+(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O)))
+(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
+O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O
+(S O) O (le_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
+Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda
+(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
+O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort
+O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)
+(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
+(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O)
+(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead
+(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort
+O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O))
+x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst)
+(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind
+Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O)
+(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T
+(TLRef (S (S O))) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef n) \Rightarrow (match n
+in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S n0)
+\Rightarrow (match n0 in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])]) | (THead _ _ _) \Rightarrow
+False])) I (TLRef (S O)) H46) in (False_ind P H47)))))) H43)))))
+(pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 x5 x1 H7)))))))
+H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) (Bind Abst) (TSort
+O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) H17))))) H14))))))))
+H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) H8)))))))
+(ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) (TSort O)
+(THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead (CHead
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
+(TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort O)
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef
+O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+definition ex2_c:
+ C
+\def
+ CSort O.
+
+definition ex2_t:
+ T
+\def
+ THead (Flat Appl) (TSort O) (TSort O).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ex2/defs.ma".
+
+include "LambdaDelta-1/nf2/defs.ma".
+
+include "LambdaDelta-1/pr2/fwd.ma".
+
+include "LambdaDelta-1/arity/fwd.ma".
+
+theorem ex2_nf2:
+ nf2 ex2_c ex2_t
+\def
+ \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O)
+(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2
+H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
+O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
+(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
+O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat
+Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O)
+(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2
+(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O)
+x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1
+(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O)
+(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t:
+T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort
+(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O))
+(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal
+T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1:
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
+(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T
+(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead
+(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2
+x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def
+(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O)
+(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O)
+x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7
+\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in
+(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr)
+(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort
+O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O)
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O)
+(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2)
+(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
+(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq
+T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0)
+x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O)
+(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead
+(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in
+(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O
+H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
+(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O))
+t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1
+x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2
+H8))))))))))))))) H1)) H0))).
+
+theorem ex2_arity:
+ \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat
+Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def
+(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda
+(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O)
+(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O)
+(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let
+H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O
+(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a
+a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P
+(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_:
+(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def
+(eq_ind A (ASort O O) (\lambda (ee: A).(match ee in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead x0 x1) H6) in (False_ind P H7))))))) H3)))))) H0))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/C/defs.ma".
+
+definition fweight:
+ C \to (T \to nat)
+\def
+ \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))).
+
+definition flt:
+ C \to (T \to (C \to (T \to Prop)))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt
+(fweight c1 t1) (fweight c2 t2))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/flt/defs.ma".
+
+include "LambdaDelta-1/C/props.ma".
+
+theorem flt_thead_sx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
+T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u)
+(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u)
+(tweight t)) (le_plus_l (tweight u) (tweight t))))))).
+
+theorem flt_thead_dx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
+T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u)
+(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u)
+(tweight t)) (le_plus_r (tweight u) (tweight t))))))).
+
+theorem flt_shift:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c
+k u) t c (THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat
+(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt
+(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus
+(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
+(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
+c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
+t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
+(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
+(tweight t))))))).
+
+theorem flt_arith0:
+ \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t
+(CHead c k t) (TLRef i)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_:
+nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
+
+theorem flt_arith1:
+ \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle
+(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i:
+nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
+\def
+ \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda
+(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_:
+K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1)
+(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H
+(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
+nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
+(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
+(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S
+O))))))))))).
+
+theorem flt_arith2:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1
+t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt
+c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda
+(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
+(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
+(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
+t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S
+O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
+
+theorem flt_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1
+t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt
+c1 t1 c3 t3))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3:
+T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1
+t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))).
+
+theorem flt_wf__q_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C
+\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq
+nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall
+(t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c:
+C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c:
+C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
+
+theorem flt_wf_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2:
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
+\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2:
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
+\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda
+(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m:
+nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda
+(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1:
+nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq
+nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0
+(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2
+(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c
+t))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubst0/defs.ma".
+
+inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def
+| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1
+t2))
+| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2
+t1))
+| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2:
+C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/fsubst0/defs.ma".
+
+theorem fsubst0_gen_base:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall
+(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1
+c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0
+i v t1 t2) (csubst0 i v c1 c2)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind
+i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v
+t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0
+i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0
+(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1
+c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0
+i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i
+v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1
+t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj
+(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0:
+T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0
+i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1
+t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj
+(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/props.ma".
+
+include "LambdaDelta-1/clear/drop.ma".
+
+theorem clear_getl_trans:
+ \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to
+(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3:
+C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1
+c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2
+c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1
+(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda
+(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to
+(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2:
+C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall
+(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda
+(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda
+(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1
+c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3)
+\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t)
+c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda
+(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to
+(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c
+(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def
+(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in
+(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3))
+(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda
+(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n
+H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f)
+t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1
+c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i).
+
+theorem getl_clear_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to
+(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1
+c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all
+c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e:
+C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1
+x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in
+(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2
+(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6
+\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5)
+in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind
+x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1
+c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0
+x1 c3 x2 H7)))))))) H4))))) H1))))))).
+
+theorem getl_clear_bind:
+ \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c
+(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2)
+\to (getl (S n) c e2))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1:
+C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2:
+C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda
+(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n)
+(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_:
+(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0)
+(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1:
+C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2:
+C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear
+(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n:
+nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0
+k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda
+(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b)
+v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1]))
+(CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1
+(Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e in
+C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _)
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b1)
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0
+(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead e1
+(Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b)
+v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8
+\def (eq_ind C e1 (\lambda (c1: C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b
+(\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b)
+n c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: F).(\lambda (H2: (clear
+(CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v
+(clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))) k
+H0))))))))))) c)).
+
+theorem getl_clear_conf:
+ \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to
+(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3:
+C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2
+c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1
+c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c:
+C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in
+(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3
+(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3
+\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead
+x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c:
+C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3
+(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall
+(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2)
+\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall
+(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n)
+c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n)
+(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0)
+c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2:
+C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda
+(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0:
+K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl
+(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b)
+t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c
+(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3
+(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4)))))
+(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda
+(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n
+H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/props.ma".
+
+theorem getl_dec:
+ \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl i c d) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P))))))
+(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P)))
+(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P:
+Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall
+(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to
+(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i:
+nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall
+(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind
+(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O
+(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl
+(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead
+c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0
+(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e:
+C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e
+(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def
+(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e
+(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to
+(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b)
+v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl
+O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O
+(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0:
+C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind
+x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat
+f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2)
+O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to
+(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v))))))
+(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))
+(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P:
+Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t
+(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n:
+nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H
+(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall
+(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t)
+(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to
+(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead
+e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0:
+C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0
+(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v))))))
+(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))
+(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n)
+(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0
+(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n)
+c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind
+b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P:
+Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t)
+d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1)))))
+i)))))) c).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/drop/defs.ma".
+
+include "LambdaDelta-1/clear/defs.ma".
+
+inductive getl (h: nat) (c1: C) (c2: C): Prop \def
+| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h
+c1 c2))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/props.ma".
+
+include "LambdaDelta-1/clear/drop.ma".
+
+theorem getl_drop:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
+nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
+C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
+T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b)
+u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n)
+e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u:
+T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0
+e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
+T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t)
+(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0:
+(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear
+(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e)))
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind
+b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow
+c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0
+(CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b |
+(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
+[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u)
+(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in
+((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
+(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e
+(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e
+c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t)
+c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0))
+(drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2))))
+(\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b)
+u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind
+b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl
+e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n:
+nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S
+n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e
+(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0:
+nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t
+n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+
+theorem getl_drop_conf_lt:
+ \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c0 e0)))))))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1:
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d))
+c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i
+(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i
+(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda
+(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i:
+nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t)
+(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def
+(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C
+(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0
+(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x:
+C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead
+c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to
+((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead
+e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))
+(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda
+(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind
+b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda
+(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u))
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda
+(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda
+(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1:
+K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0)
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0:
+B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda
+(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind
+b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0
+H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow
+(match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
+(Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0)
+(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def
+(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind
+b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0
+H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14
+\def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind
+b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i
+O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def
+(eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in
+(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead
+c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0:
+C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead
+e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))
+(\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b)
+d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20:
+(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1:
+T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1)
+H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1))
+(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind
+b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17
+e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O
+(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat
+f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n
+d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0))
+\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t)
+e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda
+(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u))
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0
+(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in
+((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1]))
+(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0
+(Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1)
+\Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead
+c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat
+f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2:
+C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b)
+u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O
+(CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def
+(eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f)
+H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d))
+(CHead c0 k1 t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0:
+C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda
+(v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2:
+T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t
+(lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f)
+(plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
+(r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2:
+C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u))
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2)
+H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda
+(t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind
+b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24
+(lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1
+(Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead
+e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))
+(\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d
+x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop
+h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0
+(Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d
+x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1
+(Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda
+(t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))
+(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h
+d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2)
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))
+x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28
+f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus
+O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi:
+(((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t)
+(CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind
+b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T
+C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus
+(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t)
+(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0)
+\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda
+(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0)
+e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
+e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0
+k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d))
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0)))
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2:
+T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r
+k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0
+x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S
+i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S
+(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0
+(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1
+(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let
+H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to
+((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
+e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda
+(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T
+t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead
+c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
+h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2)
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
+e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind
+T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k
+x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0
+O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S
+(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d))
+(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S
+i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h
+(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k
+i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0)
+(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1
+(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d
+H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4:
+C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1
+(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def
+(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S
+(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3)
+H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind
+b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22)
+in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift
+h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22))))))))
+H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k
+H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)).
+
+theorem getl_drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d
+h) i) \to (getl (minus i h) e a)))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
+d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H)
+in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0
+a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c
+x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i
+x c H3 e h d H0 H1) H4)))) H2)))))))))).
+
+theorem getl_conf_ge_drop:
+ \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i:
+nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1
+c2) \to (drop i O c2 e))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda
+(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O))
+(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e
+u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S
+i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in
+H3)))))))).
+
+theorem getl_drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
+(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i
+c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2))
+(\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
+\def
+ \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1
+e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2
+H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
+
+theorem drop_getl_trans_lt:
+ \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2:
+C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda
+(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda
+(e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
+c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i
+c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b)
+v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e:
+C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead
+e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d
+(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4:
+(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1
+e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1:
+C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1:
+C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O
+c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat
+(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i)))
+(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b
+e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h
+(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C
+(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v))))
+(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda
+(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda
+(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i
+c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h
+(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus
+d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S
+(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))).
+
+theorem drop_getl_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2
+e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 e2))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
+c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def
+(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
+(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_:
+C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i)
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x:
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def
+(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i
+O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda
+(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1:
+C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1
+e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h
+(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5)))))
+H2)))))))))).
+
+theorem drop_getl_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2)
+\to ((le d i) \to (getl (plus i h) c1 e2)))))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2:
+C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def
+(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
+(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x:
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro
+(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
+
+theorem getl_drop_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i
+h)) O c1 e2)))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h:
+nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2
+e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2:
+C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2
+H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda
+(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2:
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2
+e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall
+(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2:
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead
+c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b)
+t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop
+(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead
+c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S
+i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2))
+H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2
+(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead
+c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2
+t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_:
+(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i:
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t)
+e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S
+i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop
+(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2
+i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f:
+F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i:
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t)
+e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f)
+(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2)
+(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0)
+t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i
+n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2
+(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i)
+O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S
+(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/fwd.ma".
+
+include "LambdaDelta-1/clear/props.ma".
+
+include "LambdaDelta-1/flt/props.ma".
+
+theorem getl_flt:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to
+(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b)
+u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef
+i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u:
+T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef
+i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
+T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t)
+(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0:
+(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear
+(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef
+O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e
+(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _)
+\Rightarrow c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind
+b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b |
+(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
+[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u)
+(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in
+((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
+(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e
+(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e
+c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef
+O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef
+O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef
+O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2)))) (\lambda
+(f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b)
+u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u)
+(clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) k
+(getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n:
+nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u
+(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead
+e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind
+b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/defs.ma".
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+include "LambdaDelta-1/clear/fwd.ma".
+
+theorem getl_gen_all:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2
+C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1
+c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e:
+C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda
+(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda
+(e0: C).(clear e0 c2)) e H0 H1)))) H)))).
+
+theorem getl_gen_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h
+(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H)
+in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear
+e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2:
+(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P
+(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
+(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2
+(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0
+H1))))) H0)))))).
+
+theorem getl_gen_O:
+ \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x)))
+\def
+ \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def
+(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0))
+(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1:
+(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0
+(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))).
+
+theorem getl_gen_S:
+ \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
+nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
+nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all
+(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c
+k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0:
+C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0
+x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))).
+
+theorem getl_gen_2:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3
+B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind
+b) v)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1
+c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b:
+B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v))))))
+(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x
+c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B
+C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind
+b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4:
+(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c:
+C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind
+x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda
+(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b:
+B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c
+(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4))))))
+H3))))) H0))))).
+
+theorem getl_gen_flat:
+ \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d))))))
+\def
+ \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n
+e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e
+(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d
+H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to
+(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v)
+d)).(getl_gen_S (Flat f) e d v n H0)))) i))))).
+
+theorem getl_gen_bind:
+ \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d
+(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda
+(j: nat).(getl j e d)))))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or
+(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j:
+nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl
+O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c:
+C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda
+(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl
+(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat
+(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e
+(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind
+b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d
+(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n:
+nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n
+O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S
+j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e
+(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b)
+v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl
+j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j:
+nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n
+H0)))))) i))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/drop.ma".
+
+include "LambdaDelta-1/getl/clear.ma".
+
+theorem getl_conf_le:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
+(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e
+a))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda
+(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda
+(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e
+a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x
+e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1)
+e H4)))) H2))))))))).
+
+theorem getl_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl
+h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1
+e2)))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
+nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2
+e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e:
+C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2)
+(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x
+e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1
+e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c:
+C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O
+h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to
+(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def
+(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x
+H_y H3))))) i H2)))) H1)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/fwd.ma".
+
+include "LambdaDelta-1/drop/props.ma".
+
+include "LambdaDelta-1/clear/props.ma".
+
+theorem getl_refl:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u)
+(CHead c (Bind b) u))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind
+b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b)
+u)) (clear_bind b c u)))).
+
+theorem getl_head:
+ \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k
+h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e))))))
+\def
+ \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k
+h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0:
+C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1:
+(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k
+u) e x (drop_drop k h c x H1 u) H2)))) H0))))))).
+
+theorem getl_flat:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall
+(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c
+e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in
+(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e))
+(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c
+x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to
+(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def
+(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in
+(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c
+(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_:
+(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop
+(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat
+f) h0 c x H3 u) H2)))) h H1)))) H0))))))).
+
+theorem getl_ctail:
+ \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v:
+T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda
+(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C
+(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b)
+u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x:
+C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b)
+u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v
+x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))).
+
+theorem getl_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to
+(\forall (x2: C).((getl h c x2) \to (eq C x1 x2))))))
+\def
+ \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c
+x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all
+c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e:
+C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c
+x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in
+(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq
+C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear
+x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0
+(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0:
+C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def
+(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0
+H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+inductive iso: T \to (T \to Prop) \def
+| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2)))
+| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2)))
+| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2:
+T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/iso/defs.ma".
+
+include "LambdaDelta-1/tlist/defs.ma".
+
+theorem iso_gen_sort:
+ \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda
+(n2: nat).(eq T u2 (TSort n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1)
+u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex
+nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0:
+(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1))
+\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0:
+nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2
+\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
+with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n1) H1) in (ex_intro nat (\lambda (n3:
+nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort n2))))))) (\lambda
+(i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TSort n1))).(let
+H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex nat
+(\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1:
+T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T
+(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort n1) H1) in (False_ind (ex nat (\lambda (n2:
+nat).(eq T (THead k v2 t2) (TSort n2)))) H2)))))))) y u2 H0))) H))).
+
+theorem iso_gen_lref:
+ \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda
+(n2: nat).(eq T u2 (TLRef n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1)
+u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex
+nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0:
+(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1))
+\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0:
+nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2
+\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (TLRef n1) H1) in (False_ind (ex nat
+(\lambda (n3: nat).(eq T (TSort n2) (TLRef n3)))) H2))))) (\lambda (i1:
+nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2
+\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
+with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _)
+\Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in (ex_intro nat (\lambda (n2:
+nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))))))) (\lambda
+(v1: T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H1: (eq T (THead k v1 t1) (TLRef n1))).(let H2 \def (eq_ind T
+(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2:
+nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))).
+
+theorem iso_gen_head:
+ \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso
+(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))))))
+\def
+ \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda
+(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t:
+T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2:
+T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y
+u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to
+(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2))))))))
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k
+v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H1)
+in (False_ind (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T (TSort n2)
+(THead k v2 t2))))) H2))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda
+(H1: (eq T (TLRef i1) (THead k v1 t1))).(let H2 \def (eq_ind T (TLRef i1)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda (v2:
+T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) (\lambda
+(v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (k0:
+K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H3 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in (\lambda (_: (eq T
+v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(ex_2 T T
+(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k1 v2 t2) (THead k v3 t3))))))
+(ex_2_intro T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2)
+(THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 t2))) k0 H6)))) H3))
+H2)))))))) y u2 H0))) H))))).
+
+theorem iso_flats_lref_bind_false:
+ \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall
+(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind
+b) v t)) \to (\forall (P: Prop).P)))))))
+\def
+ \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda
+(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads
+(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))
+(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let
+H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in
+(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P
+(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let
+H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef x) H1) in
+(False_ind P H2)))) H0))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda
+(_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall
+(P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1
+(TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def
+(iso_gen_head (Flat f) t0 (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)
+H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda (v2: T).(\lambda (t2:
+T).(eq T (THead (Bind b) v t) (THead (Flat f) v2 t2)))) P (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f)
+x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat f) x0 x1) H2) in (False_ind P H3))))) H1))))))))
+vs)))))).
+
+theorem iso_flats_flat_bind_false:
+ \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall
+(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads
+(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P:
+Prop).P)))))))))
+\def
+ \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda
+(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind
+(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2))
+(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead
+(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def
+(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in
+(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t)
+(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1:
+(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T
+(THead (Bind b) v t) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))
+(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead
+(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v
+t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads
+(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1
+\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead
+(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3
+\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1))))))))
+vs)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/iso/fwd.ma".
+
+theorem iso_refl:
+ \forall (t: T).(iso t t)
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n:
+nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_:
+(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t).
+
+theorem iso_trans:
+ \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
+t3) \to (iso t1 t3)))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
+(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in
+(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3)
+(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x)
+(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1)))))))
+(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso
+(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in
+(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3)
+(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x)
+(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1)))))))
+(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def
+(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda
+(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5)
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0
+x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t))
+(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/leq/props.ma".
+
+theorem asucc_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
+(asucc g a1) (asucc g a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k)
+(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O
+(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow
+(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq
+A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n:
+nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g
+(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S
+h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k)
+(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind
+A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O
+(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq
+A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k)
+(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k))))
+(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k)
+H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g
+(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3:
+nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
+\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
+g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
+(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1)
+n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g
+(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a:
+A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3)
+n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2)
+k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort
+O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k))
+(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1))
+(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g
+(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g
+n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort
+O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A
+(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda
+(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to
+((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g
+(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow
+(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h)
+\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O
+\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))
+(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
+k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
+\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
+\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1
+(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A
+(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k))
+(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g
+(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O
+n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort
+(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k))
+(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda
+(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
+h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k))
+\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
+\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
+n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
+with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
+n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
+(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
+(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
+g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4
+n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a
+(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k))
+(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A
+(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g
+(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S
+h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
+(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k)
+(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda
+(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g
+(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
+(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
+a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
+
+theorem asucc_inj:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
+g a2)) \to (leq g a1 a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
+(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
+(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
+(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
+n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
+(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
+n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
+(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
+(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1
+g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind
+nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
+(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort
+h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
+x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2))
+(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
+\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G
+next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
+(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3:
+nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3
+O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g
+(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in
+(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a:
+A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S
+x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2))
+a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
+(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3:
+nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2)))
+\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g
+(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O
+(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
+(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
+O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda
+(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g
+(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1:
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
+x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
+\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
+nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
+n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1
+x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1
+(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
+n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4:
+nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8
+n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
+(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0)
+(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10
+(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in
+(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0))
+(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g
+(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq
+g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda
+(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to
+((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort
+n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))
+(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
+n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
+\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3
+n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
+(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
+n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2:
+nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g
+(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus
+g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
+\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G
+next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
+(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4:
+nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7)
+in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3
+n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def
+(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
+(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
+(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
+x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
+(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4:
+nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
+n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
+g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
+n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
+n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
+n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def
+(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind
+nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
+(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2
+n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0:
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g
+(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4
+n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
+\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
+nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
+n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1
+x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1
+(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0)
+x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A
+(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10
+\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
+(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
+(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2))
+a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2))
+in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1
+H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n
+n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda
+(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
+a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
+a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
+a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
+(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
+n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
+(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
+n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
+g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
+(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc
+g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g
+n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g
+(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda
+(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g
+(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1
+x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g
+(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_:
+(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a)))
+\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0)
+a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g
+(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
+n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g
+(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0)
+a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
+a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in
+(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2)
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a
+(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0))
+(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A
+(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A
+(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc
+g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1
+x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8)))))))
+H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2:
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0:
+A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to
+(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g
+(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g
+(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0))
+(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda
+(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def
+(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x
+in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda
+(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda
+(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0)
+(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a
+x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next
+g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
+(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1:
+nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0)))
+\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g
+(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a
+(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda
+(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq
+g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
+(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g
+a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def
+(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1)
+n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g
+(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda
+(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g
+(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g
+(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3
+(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5:
+A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g
+(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g
+a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g
+a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3
+(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
+((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n)
+\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h)
+\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0
+a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g
+a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def
+(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9)
+in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in
+(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2))))))
+a1)).
+
+theorem leq_asucc:
+ \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
+a0)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
+A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
+A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
+(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
+(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
+(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
+(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
+(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
+g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
+(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
+a)).
+
+theorem leq_ahead_asucc_false:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
+(asucc g a1)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
+(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
+nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
+(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
+(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1
+g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind
+A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_:
+A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
+(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda
+(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A
+(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1:
+nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O
+\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to
+P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let
+H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1
+\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S
+n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
+A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_:
+(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def
+(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H))))))
+(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g
+a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall
+(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P:
+Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2)
+(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g
+(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind
+A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_:
+A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
+(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
+x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g
+a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e
+in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc
+(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0
+with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) |
+(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead
+_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda
+(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3))
+H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g
+(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6)))))))
+H2)))))))))) a1)).
+
+theorem leq_asucc_false:
+ \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
+a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
+(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
+(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
+n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
+(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O
+(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
+O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda
+(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda
+(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
+(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A
+(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow
+n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5
+\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
+with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O
+n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r
+nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g
+(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1:
+nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7
+n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
+(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O
+n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S
+x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1:
+nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda
+(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next
+g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda
+(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g
+n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
+n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda
+(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
+(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S
+n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e
+in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _
+_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1)
+n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def
+(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g
+(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda
+(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2)))
+H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda
+(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1)
+n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g
+(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2:
+nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H)))))
+(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P:
+Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to
+(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead
+a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1)
+(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
+(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1)
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0
+x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1)
+(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _)
+\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1)
+H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2:
+A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0
+(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6)))))))
+H2))))))))) a)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/aplus/defs.ma".
+
+inductive leq (g: G): A \to (A \to Prop) \def
+| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
+(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
+| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3:
+A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2
+a4))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/leq/defs.ma".
+
+theorem leq_gen_sort1:
+ \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
+g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
+(ASort h2 n2))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
+A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda
+(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort
+h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A
+a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g
+(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat
+nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a
+k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0:
+nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
+nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
+k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal
+A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
+n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1
+n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0
+h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0
+n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
+nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
+nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda
+(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3
+n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort
+n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
+(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
+(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0)))))
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2)
+(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3
+n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
+(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0
+H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
+(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k)
+(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
+A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to
+(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
+nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2
+n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def
+(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k)
+(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0)))
+H))))).
+
+theorem leq_gen_head1:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
+(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1
+a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
+(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
+g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
+a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
+(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1
+a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
+(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq
+A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1)
+(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
+(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
+(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda
+(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2)
+(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
+(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A
+(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda
+(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4
+a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
+a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
+A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2
+a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6
+a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4)
+(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6)
+\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0
+a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
+(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
+A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
+a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6:
+A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6:
+A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
+A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda
+(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12
+\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A
+A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda
+(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5)
+(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5)))))))))
+H6))))))))))) y a H0))) H))))).
+
+theorem leq_gen_sort2:
+ \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
+g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1)
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
+(ASort h2 n2))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
+A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda
+(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
+(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq
+A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind
+g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat
+nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus
+g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0:
+nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
+nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
+k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal
+A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
+n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1
+n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2
+h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0
+n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
+nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
+nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda
+(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3
+n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort
+h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
+(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
+(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0)))))
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0)
+(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1
+n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
+(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2
+H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
+(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
+h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
+A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to
+(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
+nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2
+n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def
+(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
+(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0)))
+H))))).
+
+theorem leq_gen_head2:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a
+(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3
+a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
+(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
+g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
+a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
+(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1
+a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda
+(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq
+A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2)
+(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
+(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
+(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda
+(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1)
+(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
+(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A
+(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda
+(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4
+a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
+a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
+A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7
+a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6
+a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5)
+(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6)
+\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3
+a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
+(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_:
+A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A
+a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6:
+A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6:
+A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
+A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda
+(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12
+\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A
+A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda
+(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4)
+(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4)))))))))
+H6))))))))))) a y H0))) H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/leq/fwd.ma".
+
+include "LambdaDelta-1/aplus/props.ma".
+
+theorem ahead_inj_snd:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
+(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
+(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def
+(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A
+(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda
+(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4)
+(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead
+a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _)
+\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3)
+in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a:
+A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a:
+A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
+
+theorem leq_refl:
+ \forall (g: G).(\forall (a: A).(leq g a a))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
+(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
+(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
+a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
+H0))))) a)).
+
+theorem leq_eq:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
+a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
+a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
+
+theorem leq_sym:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
+a2 a1))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
+nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
+nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
+(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
+A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
+H1 a6 a5 H3))))))))) a1 a2 H)))).
+
+theorem leq_trans:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
+(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
+a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
+(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2
+\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3:
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3
+n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3
+(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1:
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus
+g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def
+(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1
+x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1
+n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g
+(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r
+nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n)
+(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k)
+x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus
+g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7
+H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2)
+(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus
+(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g
+(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1
+x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g
+(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3:
+A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5:
+A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6:
+A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7)
+\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6)
+a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in
+(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_:
+A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
+a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8:
+(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0
+(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead
+a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9)))))))
+H5))))))))))))) a1 a2 H)))).
+
+theorem leq_ahead_false_1:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
+n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
+(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
+(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0)
+a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4:
+A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
+(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O
+n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee:
+A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P
+H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1
+n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0)
+a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2
+(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda
+(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1)
+n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
+(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort
+(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
+(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H:
+((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
+Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
+a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
+g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def
+(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3)))
+(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda
+(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
+x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5)
+in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3]))
+(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def
+(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def
+(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0
+H10 P))))) H6))))))) H2)))))))))) a1)).
+
+theorem leq_ahead_false_2:
+ \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
+A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
+n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
+a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O
+n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
+(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
+x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0)
+(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
+(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P
+H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1
+n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1)
+n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0)
+(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
+(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1)
+n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
+a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort
+(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
+(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_:
+((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P:
+Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead
+a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq
+g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def
+(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_:
+A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4:
+A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0)
+x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5)
+in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3]))
+(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def
+(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let
+H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9
+P))))) H6))))))) H2)))))))))) a2)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/tlist/defs.ma".
+
+include "LambdaDelta-1/s/defs.ma".
+
+definition lref_map:
+ ((nat \to nat)) \to (nat \to (T \to T))
+\def
+ let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t
+with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
+(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u
+t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in
+lref_map.
+
+definition lift:
+ nat \to (nat \to (T \to T))
+\def
+ \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x:
+nat).(plus x h)) i t))).
+
+definition lifts:
+ nat \to (nat \to (TList \to TList))
+\def
+ let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with
+[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts
+h d ts0))]) in lifts.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/defs.ma".
+
+theorem lift_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
+n)) (TSort n))))
+\def
+ \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort
+n)))).
+
+theorem lift_lref_lt:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T
+(lift h d (TLRef n)) (TLRef n)))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
+d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true
+\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T
+(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
+
+theorem lift_lref_ge:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
+(lift h d (TLRef n)) (TLRef (plus n h))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
+n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
+\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h))))
+(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false
+(le_bge d n H)))))).
+
+theorem lift_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d)
+t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
+
+theorem lift_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u)
+(lift h (S d) t)))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
+
+theorem lift_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u)
+(lift h d t)))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
+
+theorem lift_gen_sort:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T
+(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind
+(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))
+(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort
+n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H:
+(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort
+n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
+(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let
+H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
+(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0)
+H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d
+n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
+(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
+(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef
+(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda
+(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq
+T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d
+t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d
+(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
+(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1))
+(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k
+t0 t1) (TSort n)) H3))))))))) t)))).
+
+theorem lift_gen_lref:
+ \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
+(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le
+(plus d h) i) (eq T t (TLRef (minus i h)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h:
+nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
+(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
+h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda
+(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort
+n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i)))
+(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1))))))))
+(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
+nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or
+(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T
+(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
+T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n)
+(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
+n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef
+n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0
+h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus
+d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n)
+(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d
+n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef
+i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h)
+(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le
+(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n
+(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n
+h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0)))))
+(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
+(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h)
+(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h))
+(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i
+H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d:
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to
+(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0
+(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d:
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to
+(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1
+(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
+nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1
+(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
+H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d)
+t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i)))
+(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h)))))
+H3)))))))))))) t).
+
+theorem lift_gen_lref_lt:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall
+(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n
+d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x
+\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d)
+(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq
+T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind
+(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n
+d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2:
+(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d
+h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus
+d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
+(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq
+T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S
+n) d h H))) t H4))) H2)) H1)))))))).
+
+theorem lift_gen_lref_false:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
+(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall
+(P: Prop).P)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
+n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T
+(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d
+h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n)))
+(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land
+(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P
+(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H
+H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n
+h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
+(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false
+(plus d h) n P H4 H0))) H3)) H2)))))))))).
+
+theorem lift_gen_lref_ge:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
+(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
+n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
+t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in
+(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus
+d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n))
+(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind
+(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda
+(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r
+T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq
+T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d
+(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2))
+(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
+h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
+h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
+(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n
+h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus
+(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
+
+theorem lift_gen_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z)))))))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
+(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda
+(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1
+\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef
+n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
+(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0:
+(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
+(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def
+(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def
+(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H
+(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead
+k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1:
+T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0
+t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2:
+T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
+(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0
+(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2]))
+(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2)
+\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1))
+H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let
+H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k
+H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z:
+T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
+k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let
+H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0:
+nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift
+h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k
+d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d)
+z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
+t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10
+(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
+t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9
+(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z))))))
+(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d)
+z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0))
+(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4))
+H3))))))))))) x)))).
+
+theorem lift_gen_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (S d) z)))))))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d
+x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
+(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
+(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d)
+x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d)
+z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro
+T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind
+b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
+z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d
+x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
+
+theorem lift_gen_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d
+x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
+(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
+(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1)
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T
+(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
+T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d
+x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
+(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1))
+(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x
+H1)))))) H0))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/fwd.ma".
+
+include "LambdaDelta-1/s/props.ma".
+
+theorem thead_x_lift_y_y:
+ \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
+(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0)
+\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n)))
+(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d
+(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n:
+nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
+(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
+(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_:
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
+h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0:
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
+h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0
+t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
+(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _)
+\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
+H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map
+(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort
+n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
+with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
+\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
+lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat
+\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
+\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
+lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow
+(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T
+\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f
+i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f
+(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec
+lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
+\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
+lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2)
+\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
+H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def
+(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall
+(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0:
+Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0
+t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0
+d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P))))))
+H3)) H2)))))))))))) t)).
+
+theorem lift_r:
+ \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0)
+t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda
+(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n))
+(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef
+n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H)))
+(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T
+t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
+(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
+t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
+t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k
+(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d)))
+(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
+
+theorem lift_lref_gt:
+ \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
+(pred n))) (TLRef n))))
+\def
+ \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef
+(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
+(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
+(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
+(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n)))
+(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
+(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
+(S_pred n d H))))))).
+
+theorem lifts_tapp:
+ \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq
+TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp
+(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp
+t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d
+t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1)
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0
+v)) H)))) vs)))).
+
+theorem lift_inj:
+ \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
+(lift h d x) (lift h d t)) \to (eq T x t)))))
+\def
+ \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
+t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
+(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
+H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
+T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
+(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
+n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
+d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
+h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
+(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
+t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
+T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
+(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1))
+\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
+nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
+t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0
+t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
+(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
+(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1
+(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
+(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
+h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
+T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
+(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b))
+(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t)
+(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t:
+T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T
+(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda
+(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d
+t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0))
+(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0))
+(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t)
+(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y:
+T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq
+T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda
+(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0)
+(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T
+(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0
+x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3))))))
+(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
+
+theorem lift_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
+nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
+((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
+t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
+d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
+x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
+(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
+(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
+(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
+(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
+d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
+(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
+(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
+(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
+(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
+h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
+d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
+(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
+(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
+T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
+(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
+n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
+(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
+(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
+(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
+(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
+(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
+(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
+T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
+(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
+n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
+t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
+h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
+(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
+(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2)))
+(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq
+T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
+(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus
+n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2)
+h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4)
+(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda
+(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2
+d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus
+n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus
+(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans
+h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T
+(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T
+t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))
+(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1
+d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n
+h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0)
+(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2)
+h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T
+(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2)))
+(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2
+n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2
+h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2)
+(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2)))))
+(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t:
+T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus
+n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat
+plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus
+(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2))
+(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2)))
+(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus
+(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4)
+(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n
+(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n
+h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
+(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift
+h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda
+(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x:
+T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2:
+nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t
+t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
+(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T
+x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2
+t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t
+t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
+(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3
+(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1))
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2
+h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2
+(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
+(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind
+b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T
+(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
+d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2))))
+(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
+t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
+x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1
+d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0)
+(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let
+H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1)
+t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2:
+T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2)
+t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift
+h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
+x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S
+d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda
+(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift
+h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead
+(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2:
+T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2
+t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2)
+(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
+(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift
+h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1
+d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
+(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
+t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)))
+(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1
+H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8))))
+(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1
+(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T
+(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def
+(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift
+h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0))
+(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z:
+T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0)
+(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x
+(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2
+h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1)
+x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda
+(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t
+t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq
+T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead
+(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0
+(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift
+h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2
+x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2
+d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2:
+T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2)))
+(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2
+t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda
+(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1
+d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2
+d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2
+t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2)
+(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f)
+(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3)
+(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2:
+T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T
+(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f)
+x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2
+x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2)
+(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2
+d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0
+H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2
+H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus
+d2 h1) H4))))) k H2))))))))))))) t1).
+
+theorem lifts_inj:
+ \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
+nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
+\def
+ \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
+d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts
+h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_:
+nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda
+(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d:
+nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t)
+(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match
+ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True |
+(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in
+(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t:
+T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h:
+nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq
+TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t
+t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d
+t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0))
+(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with
+[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in
+(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda
+(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList
+(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t
+t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList
+(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d
+t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList
+return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow
+(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda
+(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d
+t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def
+(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda
+(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat)
+(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil |
+(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in
+lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h
+d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift
+h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0)
+(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H
+t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
+
+theorem lift_free:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
+(lift h d t)) (lift (plus k h) d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
+(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
+nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T
+(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort
+n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
+(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0))
+(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k
+h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n))
+(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k:
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d
+h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef
+n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T
+(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef
+n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
+(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus
+k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1
+H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d
+n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift
+(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda
+(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n
+(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal
+nat T TLRef (plus (plus n h) k) (plus n (plus k h))
+(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
+(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
+(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1
+(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0:
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
+(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda
+(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d:
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e
+(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda
+(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d
+h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k
+d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0
+t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift
+h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0
+t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d)
+t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k
+e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h
+d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift
+(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d)
+(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le
+k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
+(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e
+(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift
+h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h
+d))))))))))))) t).
+
+theorem lift_d:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t))
+(lift k e (lift h d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
+d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda
+(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_:
+(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0)
+(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
+T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0:
+T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
+T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k
+e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n))
+(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
+nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h
+(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda
+(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef
+n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d
+(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k
+e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift
+h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n))
+(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k
+d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e
+n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d)
+t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0:
+nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n)))))
+(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d
+(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda
+(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef
+(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
+(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
+(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
+(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1:
+(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0
+(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda
+(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef
+(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0))
+(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat
+(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k
+e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0)))
+(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus
+n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n
+k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e
+H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h:
+nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq
+T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d
+t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0:
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0
+d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h:
+nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le
+e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
+T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
+(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
+k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d
+(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1))
+(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h
+(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead
+k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda
+(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
+k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda
+(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift
+k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h
+(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e
+t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e)
+t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1)
+(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0
+d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k
+(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1))
+(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0
+(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0
+d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/fwd.ma".
+
+include "LambdaDelta-1/tlt/props.ma".
+
+theorem lift_weight_map:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
+(weight_map f (lift h d t)) (weight_map f t))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
+(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0)))))))
+(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m)
+O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m)
+O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f
+(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat
+(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0)))
+(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq
+nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda
+(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0))
+(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
+(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f
+t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
+(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f
+t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m)
+O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
+t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead
+(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat
+(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda
+(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift
+h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d)
+t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map
+(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f
+(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with
+[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S
+(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0)
+(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0)
+(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n:
+nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S
+(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))))
+(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n:
+nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0)
+(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus
+(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))
+(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d)
+(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d)
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
+n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda
+(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda
+(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3))))
+(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r
+nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map
+f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O)
+t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
+(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2
+nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map
+(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat
+(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h
+(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat
+(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd
+f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le
+d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x
+H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1)
+(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus
+(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus
+(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f
+t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f
+(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd
+f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1))))
+(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m:
+nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
+n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x:
+nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d
+m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h
+d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s
+(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f
+(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d
+t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1))
+(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0)
+(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1)))
+(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
+k)))))))))) t).
+
+theorem lift_weight:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d
+t)) (weight t))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d
+(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat
+O)))))).
+
+theorem lift_weight_add:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to
+(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
+(weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h:
+nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat
+(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))
+\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
+t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m:
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d)
+w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
+m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n))))))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m
+d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1:
+((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d
+(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d
+(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n))
+(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef
+n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d
+H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
+(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f
+(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda
+(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f
+(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h))
+(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift
+h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda
+(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to
+(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d
+m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0))
+(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0:
+((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f
+m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g
+(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift
+(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m:
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d)
+w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
+m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
+t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b:
+B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1))
+(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead
+(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h)
+(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b)
+(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind
+(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus
+(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d
+t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h
+d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S
+(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
+t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h)
+d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h)
+(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0))
+(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
+t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
+(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift
+(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map
+f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S
+(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S
+(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2
+H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S
+(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d
+t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m
+O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift
+(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat
+nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq
+nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g
+H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S
+m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat
+m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g
+(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda
+(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r
+nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d
+t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6))))
+H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
+n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x:
+nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0)))
+n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus
+(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
+t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g
+(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map
+(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O)
+(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O)
+(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))
+(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat
+O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m
+H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda
+(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0)))
+(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x:
+nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6))))
+H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
+n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S
+x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g
+n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat
+S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
+t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S
+h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0))
+(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1))
+(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S
+d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda
+(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n)
+(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0:
+nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda
+(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O
+m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda
+(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n)
+(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m:
+nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
+n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x:
+nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5))))
+(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head
+(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind
+b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0)
+(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2)
+(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead
+(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2:
+T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0)
+d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d
+t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0))
+(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f
+(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1))
+(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3)))
+(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d))
+(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
+k))))))))))))) t)).
+
+theorem lift_weight_add_O:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to
+nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h)
+O t))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to
+nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m:
+nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m)))))
+(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal
+nat (f m)))))))).
+
+theorem lift_tlt_dx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(tlt t (THead k u (lift h d t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight
+(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t)
+(lift_weight t h d)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/defs.ma".
+
+definition trans:
+ PList \to (nat \to nat)
+\def
+ let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i:
+nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let
+j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false
+\Rightarrow (plus j h)]))])) in trans.
+
+definition lift1:
+ PList \to (T \to T)
+\def
+ let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds
+with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0
+t))])) in lift1.
+
+definition lifts1:
+ PList \to (TList \to TList)
+\def
+ let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with
+[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t)
+(lifts1 hds ts0))]) in lifts1.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift1/defs.ma".
+
+include "LambdaDelta-1/lift/fwd.ma".
+
+theorem lift1_sort:
+ \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
+\def
+ \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T
+(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p
+(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0
+n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
+
+theorem lift1_lref:
+ \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef
+(trans hds i))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T
+(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T
+(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p
+i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq
+T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow
+(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T
+(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false
+\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
+
+theorem lift1_bind:
+ \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss
+hds) t))))))
+\def
+ \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal
+T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
+(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda
+(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p)
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p
+u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n
+n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0
+(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t)))))
+(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1
+(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t)))
+(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u
+t)) (H u t)))))))) hds)).
+
+theorem lift1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds
+t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f)
+(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T
+(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
+(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u:
+T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t))
+(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u))
+(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p
+u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift
+n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f)
+(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f)
+(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1
+p (THead (Flat f) u t)) (H u t)))))))) hds)).
+
+theorem lift1_cons_tail:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq
+T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t)
+(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1
+(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d
+t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p
+h d) t) H))))) hds)))).
+
+theorem lifts1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts:
+TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds
+ts) (lift1 hds t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts:
+TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0
+t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1
+hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds
+(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds
+t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f)
+t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads
+(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f)
+(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1
+hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1)
+(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat
+f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H)
+(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0
+(THeads (Flat f) t1 t)))))) ts)))).
+
+theorem lifts1_nil:
+ \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
+\def
+ \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t)
+t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H:
+(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq
+TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1
+PNil t0) H)))) ts).
+
+theorem lifts1_cons:
+ \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts:
+TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts:
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t)
+(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d
+(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1:
+TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1
+hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d
+(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0)
+H)))) ts)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/props.ma".
+
+include "LambdaDelta-1/drop1/defs.ma".
+
+theorem lift1_lift1:
+ \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1
+(lift1 is2 t)) (lift1 (papp is1 is2) t))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2:
+PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2)
+t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H:
+((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1
+(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat
+T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal
+nat n) (refl_equal nat n0) (H is2 t)))))))) is1).
+
+theorem lift1_xhg:
+ \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t))
+(lift (S O) O (lift1 hds t))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T
+(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t:
+T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p)
+(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T
+(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S
+O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n:
+nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d
+(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda
+(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift
+(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1
+p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S
+d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
+
+theorem lifts1_xhg:
+ \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts
+(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
+\def
+ \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq
+TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t))))
+(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq
+TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds
+t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList
+(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1
+hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O
+(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1
+hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds
+t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
+(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds)
+(lift (S O) O t)) (lift1_xhg hds t))))) ts)).
+
+theorem lift1_free:
+ \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds
+(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i:
+nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O
+(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T
+(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0:
+PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0
+(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i)
+t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0
+i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift
+(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) |
+false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0
+i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans
+hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt
+(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i))
+O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans
+hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with
+[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) |
+false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind
+(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d
+(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
+h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))))
+(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S
+(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h
+n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t))))
+(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i)))
+(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t))))
+(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans
+hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S
+(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t)))
+(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans
+hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S
+(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d
+(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0:
+(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans
+hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S
+(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus
+h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i)
+t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t))))
+(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O
+(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans
+hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1
+(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i)))
+(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S
+(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0
+i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i)))
+(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda
+(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d
+(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i))
+(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans
+hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t))))))))
+hds).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/A/defs.ma".
+
+definition lweight:
+ A \to nat
+\def
+ let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow
+O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in
+lweight.
+
+definition llt:
+ A \to (A \to Prop)
+\def
+ \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/llt/defs.ma".
+
+include "LambdaDelta-1/leq/defs.ma".
+
+theorem lweight_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
+(lweight a1) (lweight a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3:
+A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight
+a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda
+(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight
+a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
+(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2
+H)))).
+
+theorem llt_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
+(a3: A).((llt a1 a3) \to (llt a2 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1
+\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0
+(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
+
+theorem llt_trans:
+ \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2
+a3) \to (llt a1 a3)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight
+a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans
+(lweight a1) (lweight a2) (lweight a3) H H0))))).
+
+theorem llt_head_sx:
+ \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1)
+(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))).
+
+theorem llt_head_dx:
+ \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1)
+(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))).
+
+theorem llt_wf__q_ind:
+ \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to
+Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0
+a))))) P n))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
+n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight
+a)))))).
+
+theorem llt_wf_ind:
+ \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1
+a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
+(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind
+(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0:
+A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
+(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
+(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P
+a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt
+(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight
+a1))))))))))))) a)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/G/defs.ma".
+
+definition next_plus:
+ G \to (nat \to (nat \to nat))
+\def
+ let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O
+\Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/next_plus/defs.ma".
+
+theorem next_plus_assoc:
+ \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq
+nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0:
+nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2)
+(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus
+g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat
+(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda
+(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus
+g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda
+(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1))))
+(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0))
+(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n
+n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1))
+(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0))
+n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g
+(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0)
+(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))).
+
+theorem next_plus_next:
+ \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g
+(next g n) h) (next g (next_plus g n h)))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus
+g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n
+h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n
+(S O)) h) (next_plus_assoc g n (S O) h)))).
+
+theorem next_plus_lt:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next
+g n) h))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0:
+nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n))
+(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next
+g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0))
+n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next
+g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next
+g n0) n)) (next_plus_next g (next g n0) n))))) h)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/nf2/fwd.ma".
+
+include "LambdaDelta-1/arity/subst0.ma".
+
+theorem arity_nf2_inv_all:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat
+(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
+(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda
+(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w:
+T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w:
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
+c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort
+n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort
+n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n
+(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u)
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w:
+T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n:
+nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0:
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0:
+nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef
+i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
+(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
+Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads
+(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
+ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
+i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u
+(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef
+i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda
+(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
+(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i)
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
+(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
+i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
+(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
+i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
+nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
+(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0
+(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0
+(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind
+b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2
+(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead
+c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u
+t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0
+(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
+nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
+(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq
+B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0
+a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0
+H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr)
+u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
+w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
+(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g
+(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u
+t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False return (\lambda
+(_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
+(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
+(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void
+Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8:
+(nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0
+(Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda
+(v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
+(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
+Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda
+(H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1:
+T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T
+(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
+(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
+Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11
+(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u
+(lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O
+x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
+T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3
+H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T
+T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0))))
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
+T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
+(Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to
+(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst)
+w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w)))
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind
+Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList
+nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind
+Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind
+Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u
+t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2
+(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0:
+T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u
+t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
+T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2
+(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
+(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
+Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w:
+T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind
+Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
+nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
+(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
+t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def
+(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0)
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
+(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList
+nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0)
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def
+(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
+w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w:
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
+(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9:
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w:
+T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
+c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
+T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
+t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
+T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10:
+(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_:
+(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1:
+T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in
+(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2
+(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
+(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
+(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
+Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1)
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind
+Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind
+Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat
+Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10))))))))
+H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind
+nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w:
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
+(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x:
+nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda
+(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def
+(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x)
+H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w:
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
+(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def
+(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12))
+in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq
+g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
+A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead
+(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
+(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
+(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x))
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
+x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0
+x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead x0 x1) H16) in (False_ind (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead
+(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
+(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
+(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x))
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) H17))))))) H13))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
+(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0:
+TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0
+(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef
+x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl)
+u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T
+t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0
+(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda
+(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
+Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
+t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
+(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0))))
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
+T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
+(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
+(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads
+(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda
+(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u
+(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11)
+H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda
+(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
+(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda
+(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
+T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0
+u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
+Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0
+t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
+Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
+T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
+nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda
+(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
+u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
+Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
+T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind
+Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind
+Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3
+(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
+u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead
+(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
+u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))))
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
+Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w:
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
+c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8))
+t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort
+n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T
+(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u:
+T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2
+c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))))
+(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T
+(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u))))
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
+T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort
+x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x
+(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
+T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w:
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
+c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
+ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda
+(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0
+(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef
+x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3
+(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
+u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
+(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads
+(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w:
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
+c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl)
+x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
+(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
+x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0
+H6)))))) H5)) H4))))))))))) c t a H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/nf2/defs.ma".
+
+include "LambdaDelta-1/pr2/clen.ma".
+
+include "LambdaDelta-1/pr2/fwd.ma".
+
+include "LambdaDelta-1/pr0/dec.ma".
+
+include "LambdaDelta-1/C/props.ma".
+
+theorem nf2_dec:
+ \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq
+T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
+\def
+ \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall
+(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda
+(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in
+(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to
+(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2
+(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2
+H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to
+(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2:
+T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T
+t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))
+(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x
+H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or
+(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1
+t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H
+t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0)
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1:
+((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0:
+K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2
+T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0:
+B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1
+t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def
+(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v:
+T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O)
+(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2)
+\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda
+(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq
+T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O)
+(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2
+(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
+(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O)
+(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def
+H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t
+c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind
+Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2)
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1
+t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t
+c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2
+(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
+(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1
+t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0)
+x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0
+(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in
+(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0)
+(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt
+(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym
+(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t
+(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5))))
+(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1
+(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1
+(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda
+(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T
+t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall
+(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T
+(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O)
+(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
+(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda
+(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1
+\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let
+H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda
+(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O)
+(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift
+(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x)
+t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
+Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0:
+T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr)
+(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda
+(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2)
+(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10:
+(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0
+t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3)))
+(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1:
+T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x
+x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2))
+H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O)
+(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0))
+(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen
+c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x)
+t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4)))
+H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2)
+\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda
+(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def
+(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
+(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr)))
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
+(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
+K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
+(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee:
+K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow
+(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow
+False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))
+(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2:
+T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def
+(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
+(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr)))
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
+(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
+K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
+(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee:
+K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow
+(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow
+False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))
+b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0)
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda
+(t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def
+(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2
+c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2)
+(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_:
+T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f)
+(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen
+c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f)
+(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0
+t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))))
+k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))
+(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x)
+\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall
+(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t
+c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1
+x H3 k t)))))) H1)) H0)))))))) c).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr2/defs.ma".
+
+definition nf2:
+ C \to (T \to Prop)
+\def
+ \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1
+t2)))).
+
+definition nfs2:
+ C \to (TList \to Prop)
+\def
+ let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil
+\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))])
+in nfs2.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/nf2/defs.ma".
+
+include "LambdaDelta-1/pr2/clen.ma".
+
+include "LambdaDelta-1/subst0/dec.ma".
+
+include "LambdaDelta-1/T/props.ma".
+
+theorem nf2_gen_lref:
+ \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2
+c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P:
+Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0
+(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef
+i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
+
+theorem nf2_gen_abst:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u
+t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
+T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t)
+t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2:
+T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2:
+T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
+(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst)
+u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2
+H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u
+t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u)
+t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t
+t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
+_ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H
+(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in
+H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind
+Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0))
+(refl_equal T t) t2 H1))))))))).
+
+theorem nf2_gen_cast:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u
+t)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
+(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t
+(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
+
+theorem nf2_gen_beta:
+ \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
+(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
+((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2)
+\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P:
+Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind
+Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead
+(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in
+(False_ind P H0))))))).
+
+theorem nf2_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
+(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f)
+u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall
+(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c
+u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t)
+(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1)))
+(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
+(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
+(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
+
+theorem nf2_gen__nf2_gen_aux:
+ \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
+(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u:
+T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
+(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
+nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort
+n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
+d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n:
+nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u
+(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind
+T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
+(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall
+(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
+(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u:
+T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to
+(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1:
+(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
+t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
+_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
+(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T
+T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1]))
+(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let
+H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat)))
+(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort
+n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
+| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0
+(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0:
+nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat)
+(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) |
+(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map
+f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus
+x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map
+(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort
+n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
+with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
+\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f:
+((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
+\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1)
+\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
+t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0))
+H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t
+t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
+(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8
+P)))))) H3)) H2))))))))))) x)).
+
+theorem nf2_gen_abbr:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u
+t)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
+T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t)
+t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x
+in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t
+(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift
+(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O
+x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O
+x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
+_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S
+O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind
+Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u)
+t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda
+(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in
+(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O)
+O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c
+(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H
+(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c
+(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x
+(pr0_refl x) u))) P))) H1))) H0))))))).
+
+theorem nf2_gen_void:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
+(lift (S O) O t))) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
+T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
+Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux
+Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t
+(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u)))
+P))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/nf2/pr3.ma".
+
+include "LambdaDelta-1/pr3/fwd.ma".
+
+include "LambdaDelta-1/iso/props.ma".
+
+theorem nf2_iso_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
+TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso
+(THeads (Flat Appl) vs (TLRef i)) u))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads
+(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u))))
+(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def
+(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda
+(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda
+(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t:
+T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i))
+u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat
+Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl)
+t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
+T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2:
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
+i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u
+(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat
+Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0
+x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0
+(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0
+(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda
+(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
+T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1
+t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u)
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
+(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda
+(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0
+x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
+u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in
+(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2:
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
+u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift
+(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1
+x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0
+(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0
+H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
+u))))))))))))))) H3)) H2))))))) vs)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/nf2/props.ma".
+
+include "LambdaDelta-1/drop1/fwd.ma".
+
+theorem nf2_lift1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1
+hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p
+t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c
+e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in
+(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c:
+C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p
+t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p)
+c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0)
+in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda
+(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x:
+C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x
+(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/nf2/defs.ma".
+
+include "LambdaDelta-1/pr3/pr3.ma".
+
+theorem nf2_pr3_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c
+t1) \to (eq T t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t
+t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t
+(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0)
+\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def
+(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def
+(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T
+t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
+
+theorem nf2_pr3_confluence:
+ \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2)
+\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2:
+T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t
+t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0))
+(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3:
+(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1
+x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1
+H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y)
+in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2
+(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0:
+T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2
+t1 H1))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/nf2/defs.ma".
+
+include "LambdaDelta-1/pr2/fwd.ma".
+
+theorem nf2_sort:
+ \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
+\def
+ \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort
+n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal
+T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
+
+theorem nf2_csort_lref:
+ \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i)))
+\def
+ \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort
+n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq
+T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n)
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S
+i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r
+T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2
+H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort
+n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift
+(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort
+n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift
+(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2
+(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T
+(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i)
+(lift (S i) O x1))) t2 H3))))) H1)) H0))))).
+
+theorem nf2_abst:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v:
+T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind
+Abst) u t))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
+\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda
+(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t
+t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t)
+t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead
+(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2
+(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5:
+((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t
+x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead
+(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t
+x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3))))))
+H2)))))))))).
+
+theorem nf2_abst_shift:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c
+(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
+\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2
+(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda
+(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2
+H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2
+c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
+b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T
+(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst)
+u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2
+H3)))))) H2)))))))).
+
+theorem nfs2_tapp:
+ \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t))
+\to (land (nfs2 c ts) (nf2 c t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0:
+TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H:
+(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True
+(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I
+H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c
+(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c
+t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c
+(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2:
+(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let
+H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c
+t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj
+(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5)
+H6))) H4))))) H1)))))) ts))).
+
+theorem nf2_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
+TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads
+(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0
+(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in
+(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat
+Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c
+t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def
+(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
+t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t
+x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T
+(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1:
+T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0
+(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i))
+(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
+i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1:
+T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1
+(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2
+H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst)
+x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2
+c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
+u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind
+(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T
+(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead
+(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2
+x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda
+(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0
+x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
+x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil
+(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3:
+TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T
+(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead
+(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2
+x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef
+i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i))
+(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1
+(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat
+Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2
+x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0:
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10:
+(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
+x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl)
+t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1:
+TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat
+Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t
+(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl)
+(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil
+(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead
+(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
+(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat
+Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
+x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c
+(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef
+i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat
+Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
+x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef
+i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i))
+(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1
+(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat
+Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5
+(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2
+H10))))))))))))) H7)) H6))))))) H2)))))) vs)))).
+
+theorem nf2_appl_lref:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c
+(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i:
+nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0
+(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))).
+
+theorem nf2_lref_abst:
+ \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c
+(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c
+(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2
+(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d
+(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O
+u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T
+(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2
+H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c
+(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift
+(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c
+(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift
+(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i)
+O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t))
+(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c
+c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H
+(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst)
+u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort
+_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind
+Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1)
+H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2))
+H1)))))))).
+
+theorem nf2_lift:
+ \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2)
+\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i:
+nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c
+(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind
+T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3))
+(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i
+x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq
+T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x
+(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq
+T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3))))
+H2)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr1/defs.ma".
+
+definition pc1:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda
+(t: T).(pr1 t2 t)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc1/defs.ma".
+
+include "LambdaDelta-1/pr1/pr1.ma".
+
+theorem pc1_pr0_r:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H)
+(pr1_refl t2)))).
+
+theorem pc1_pr0_x:
+ \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1)
+(pr1_pr0 t2 t1 H)))).
+
+theorem pc1_refl:
+ \forall (t: T).(pc1 t t)
+\def
+ \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0:
+T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
+
+theorem pc1_pr0_u:
+ \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3:
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t:
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x:
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda
+(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2)
+H3)))) H1)))))).
+
+theorem pc1_s:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2
+t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2
+x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2
+H1)))) H0)))).
+
+theorem pc1_head_1:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall
+(k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t:
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0))
+(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda
+(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T
+(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2
+t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k)))))
+H0)))))).
+
+theorem pc1_head_2:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall
+(k: K).(pc1 (THead k u t1) (THead k u t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u:
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t))
+(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x:
+T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda
+(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead
+k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))).
+
+theorem pc1_t:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3:
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t:
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x:
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1
+t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2
+x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1
+t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x
+x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1
+(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x
+H2))))) H4))))) H1)))))).
+
+theorem pc1_pr0_u2:
+ \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0
+t2) \to (pc1 t1 t2)))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2:
+T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))).
+
+theorem pc1_head:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall
+(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2
+t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead
+k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2)
+(pc1_head_2 t1 t2 H0 u2 k)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/arity_props.ma".
+
+include "LambdaDelta-1/nf2/fwd.ma".
+
+theorem pc3_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c
+u1 u2) ((pc3 c u1 u2) \to False)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2
+t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2)
+((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1
+x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4
+\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c
+u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda
+(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0)
+in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P:
+Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x
+x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let
+H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl
+(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10)))))
+(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1
+u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9
+in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t))
+False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2
+x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2
+in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t))
+False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1
+x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def
+(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def
+(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0
+H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))).
+
+theorem pc3_abst_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to False))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2
+H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T
+(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c
+x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T
+(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T
+(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2
+c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T
+(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x
+(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda
+(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u:
+T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_:
+T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c
+v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False)))
+(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0
+t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or
+(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2
+u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u)
+t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_:
+T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind
+Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead
+(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5
+(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t:
+T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def
+(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead
+(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind
+Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda
+(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst)
+x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1
+x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda
+(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P:
+Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1
+(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead
+(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2)))
+(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1
+(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c
+u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t:
+T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False
+(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead
+(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t:
+T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda
+(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17
+\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def
+(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda
+(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20:
+(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c
+(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t
+x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4
+x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead
+c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda
+(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3
+x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
+u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0:
+T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10
+(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t:
+T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def
+(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst)
+x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind
+Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in
+(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6)
+(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K
+(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False))))
+H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4)))))
+H12)))))) H9)))))) H6)))))) H3)))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/defs.ma".
+
+definition pc3:
+ C \to (T \to (T \to Prop))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3
+c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
+
+inductive pc3_left (c: C): T \to (T \to Prop) \def
+| pc3_left_r: \forall (t: T).(pc3_left c t t)
+| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
+| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/left.ma".
+
+include "LambdaDelta-1/fsubst0/defs.ma".
+
+include "LambdaDelta-1/csubst0/getl.ma".
+
+theorem pc3_pr2_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1
+t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i:
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3
+t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0:
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0:
+T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u))
+\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr)
+u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2:
+T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c
+t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4
+w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0
+t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x:
+T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x
+t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3
+(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl
+u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e:
+C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3
+(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2
+t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e:
+C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2
+T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0
+t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3
+H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2:
+T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda
+(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0
+t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x
+H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c
+c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5))
+(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1))))))))))
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr)
+u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0
+H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0
+t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8:
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11:
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
+(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u
+H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
+(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda
+(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3
+t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
+t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
+u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3
+u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
+x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
+u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let
+H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5)))
+H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
+i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0
+(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8))
+(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
+x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
+i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda
+(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
+t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
+u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4
+u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6:
+(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c
+c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5:
+T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5:
+(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind
+Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8
+\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in
+(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b)
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))
+(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2
+c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5
+(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2
+t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1
+(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda
+(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow
+d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind
+x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
+(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda
+(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0
+t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5
+(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2
+(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0
+e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
+H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0
+(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b)
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
+x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
+u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let
+H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6)))
+H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
+i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
+(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0
+(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9))
+(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
+x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in
+(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def
+(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u
+H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
+i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda
+(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H22: (subst0 i x4
+t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0
+t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5
+(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2
+(pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0
+e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
+H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back t3 t0 u i H2 x4 u0
+(minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) (\lambda (H7:
+(le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0
+(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4)
+t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0
+H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4
+H3)))))))))))))))) c1 t1 t H)))).
+
+theorem pc3_pr2_fsubst0_back:
+ \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
+c2 t t2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t
+t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i:
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0
+t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0:
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0:
+T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u))
+\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr)
+u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4
+(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0:
+C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c
+(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0))))))
+(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda
+(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e
+(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0
+t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e
+(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1))))))))))
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr)
+u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3
+H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3
+(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0
+(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8:
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11:
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
+(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u
+H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
+(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda
+(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3
+t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
+t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
+u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3
+u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
+x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
+u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let
+H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5)))
+H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
+i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0
+(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8))
+(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
+x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
+i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda
+(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
+t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
+u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4
+u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6:
+(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c
+c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5:
+T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5:
+(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind
+Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8
+\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in
+(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b)
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))
+(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u
+c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3
+(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C
+T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1
+(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda
+(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow
+d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind
+x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
+(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda
+(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x
+t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr)
+u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0
+i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6)
+t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0
+(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b)
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
+x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
+u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let
+H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6)))
+H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
+i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2
+(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0
+e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
+H6) t0 t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9:
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b)
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
+x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13:
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in
+(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def
+(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u
+H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
+i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda
+(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
+i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4
+t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
+H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
+nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x
+t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3
+t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8)))
+(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i
+(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0
+H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n
+i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5
+H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))).
+
+theorem pc3_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1
+t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i:
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2
+t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3
+t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2:
+C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u
+c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e
+(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0
+i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr)
+u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3
+H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e:
+C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0)))))
+(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda
+(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e
+(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i
+(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3
+H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0:
+(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2:
+((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4:
+T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind
+Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u:
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2
+t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e:
+C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5:
+T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1
+(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1
+t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e
+(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0
+(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2
+c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda
+(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6:
+(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0
+t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2
+(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda
+(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u:
+T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall
+(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4
+t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda
+(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u))
+\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2
+t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr)
+u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1
+t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e
+(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0
+t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0
+(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4:
+(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1
+c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr)
+u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0
+t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst
+i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/props.ma".
+
+include "LambdaDelta-1/pr3/fwd.ma".
+
+theorem pc3_gen_sort:
+ \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort
+n)) \to (eq nat m n))))
+\def
+ \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c
+(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c
+(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x:
+T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let
+H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n
+H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0)
+\Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m]))
+(TSort m) (TSort n) H3) in H4))))) H0))))).
+
+theorem pc3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall
+(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
+t1 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
+t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst)
+u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c
+u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))))
+(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda
+(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2
+x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u:
+T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2
+x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3)))))
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
+t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead
+(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def
+(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead
+(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _)
+\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3)
+(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 |
+(TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst)
+x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let
+H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2
+(\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b:
+B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16
+u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1
+(H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))).
+
+theorem pc3_gen_abst_shift:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c
+(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind
+Abst) u) t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def
+(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u)
+(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3
+(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2:
+((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2
+Abst u))) H0))))))).
+
+theorem pc3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall
+(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d
+c e) \to (pc3 e t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e:
+C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t:
+T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e
+t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3:
+(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in
+(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e
+t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d
+x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e
+H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
+T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift
+h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t:
+T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1
+(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e
+t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
+
+theorem pc3_gen_not_abst:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1:
+T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b)
+u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S
+O) O (THead (Bind Abst) u2 t2))))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
+(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2:
+T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead
+c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda
+(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr)
+u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t:
+T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind
+Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind
+Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1)
+x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
+(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr)
+u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead
+c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5:
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
+(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1)
+t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1
+x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def
+(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
+c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
+T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1
+(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2
+x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
+u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind
+Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T
+(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
+H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1
+(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
+t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2
+t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
+Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0:
+B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def
+(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O
+t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1)
+t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind
+Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind
+Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0
+x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6)))
+H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
+t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda
+(_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst)
+u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void
+Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead
+(Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c
+(THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2
+t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2
+t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1)
+x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
+(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall
+(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1)
+t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead
+(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
+c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
+T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c
+(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0
+x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u:
+T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x
+H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
+u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind
+Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x
+(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_:
+((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let
+H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6
+(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst)
+x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0)
+\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3
+(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
+H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1
+(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
+t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2
+t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
+Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0:
+B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def
+(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O
+t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1)
+t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind
+Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind
+Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0
+x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6)))
+H4))))) H1))))))))) b).
+
+theorem pc3_gen_lift_abst:
+ \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall
+(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2))
+\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda
+(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
+t1)))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind
+Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H
+in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3
+c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1:
+T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1)))))))
+(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c
+(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0)
+in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3
+e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind
+Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1))))
+(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T
+x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2
+t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e
+t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2
+(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1
+x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x
+(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1:
+T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1:
+T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
+t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead
+(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14:
+(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10
+(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c
+u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0:
+T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T
+(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1))))
+(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_:
+T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1
+x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
+
+theorem pc3_gen_sort_abst:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c
+(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda
+(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0
+\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0:
+T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c
+(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def
+(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
+T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u
+x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
+u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n)))
+(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind
+T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P
+H8)))))))) H3))))) H0))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/props.ma".
+
+theorem pc3_ind_left__pc3_left_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda
+(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2
+c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2:
+(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_trans:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
+(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3:
+T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda
+(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
+t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0
+t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0:
+(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda
+(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0
+t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_sym:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
+(pc3_left c t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t)))
+(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
+(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
+t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3
+H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4
+t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3
+(pc3_left_r c t3))))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_pc3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x
+(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x
+(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
+
+theorem pc3_ind_left__pc3_pc3_left:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
+(pc3 c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda
+(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c
+t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3
+c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3
+(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
+
+theorem pc3_ind_left:
+ \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t
+t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
+T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1:
+T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to
+((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c
+t t0) \to (P t t0))))))))
+\def
+ \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall
+(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1
+t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1
+t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
+\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2
+t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t
+t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3:
+T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3
+(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left
+c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3
+(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0
+(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/defs.ma".
+
+include "LambdaDelta-1/nf2/pr3.ma".
+
+theorem pc3_nf2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
+t1) \to ((nf2 c t2) \to (eq T t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in
+(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T
+t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2
+x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x
+(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x
+(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c
+t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1
+H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2
+H_y0))))))))) H2))))))).
+
+theorem pc3_nf2_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
+t2) \to (pr3 c t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t:
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x:
+T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def
+(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t:
+T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/defs.ma".
+
+include "LambdaDelta-1/pc1/defs.ma".
+
+include "LambdaDelta-1/pr3/pr1.ma".
+
+theorem pc3_pc1:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c:
+C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t:
+T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1
+x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t))
+(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c)))))
+H0))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/defs.ma".
+
+include "LambdaDelta-1/pr3/pr3.ma".
+
+theorem clear_pc3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind
+T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2
+t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1
+H0))))) H1))))))).
+
+theorem pc3_pr2_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
+
+theorem pc3_pr2_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
+
+theorem pc3_pr3_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 H (pr3_refl c t2))))).
+
+theorem pc3_pr3_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) H)))).
+
+theorem pc3_pr3_t:
+ \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall
+(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1
+t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t:
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
+
+theorem pc3_refl:
+ \forall (c: C).(\forall (t: T).(pc3 c t t))
+\def
+ \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0))
+(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
+
+theorem pc3_s:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c
+t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t))
+(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
+
+theorem pc3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u
+t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead
+(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead
+(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f)))))
+H0))))))).
+
+theorem pc3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda
+(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t)
+(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2:
+(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0))
+(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x
+H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl
+(CHead c k x) t)))))) H0))))))).
+
+theorem pc3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T
+(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u)
+t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1:
+(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2
+T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u
+t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1)
+(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
+
+theorem pc3_pr2_u:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t))
+x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
+
+theorem pc3_t:
+ \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1
+x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t))
+(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7:
+(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c
+H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2)))))
+H4))))) H1))))))).
+
+theorem pc3_pr2_u2:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
+(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
+t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x
+c t1 t0 H) t2 H0)))))).
+
+theorem pc3_pr3_conf:
+ \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall
+(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t
+t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c
+t2 t H0) t1 H)))))).
+
+theorem pc3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
+(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c
+u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pc3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
+(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c
+u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
+
+theorem pc3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
+t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3:
+(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0
+(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k
+u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k
+u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in
+(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr)
+u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k
+u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1)
+t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr)
+u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead
+c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda
+(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr)
+u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d
+(Bind Abbr) u) u2 H10)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match
+e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _
+k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in
+((let H13 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead
+d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind
+Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d
+c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13)
+in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t))
+(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0))
+(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O
+u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x
+(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c
+u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr)
+u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2
+t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10:
+(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans
+(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t
+(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9))
+(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f
+c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2)
+(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0
+(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3
+(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d
+(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0:
+K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4
+t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind
+Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_:
+(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
+t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind
+b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t
+(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d
+(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_:
+(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
+t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat
+f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t
+(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f
+u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7
+H4)))))))))))))) y t1 t2 H1))) H0)))))))).
+
+theorem pc3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2
+u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3
+(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
+K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
+(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1:
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y:
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4
+t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
+c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n:
+nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
+t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3
+(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
+(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0
+| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind
+b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1)
+(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
+Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr)
+u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let
+H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in
+(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind
+T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3
+(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2
+t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5
+t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind
+Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda
+(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus
+i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat
+(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in
+(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t)
+c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind
+Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind
+Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24))))))))
+(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O
+H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear
+(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0
+(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta
+(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11))
+(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0)
+(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O
+(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda
+(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0
+t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead
+c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5
+t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0))
+\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r
+(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t)
+t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0
+(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda
+(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead
+c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1)
+H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1
+i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1
+H)))).
+
+theorem pc3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to
+(pc3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
+u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1)
+\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2
+u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
+u1 H3)))))))))) t1 t2 H)))))).
+
+theorem pc3_pr3_pc3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2
+u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
+t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3
+(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0:
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0
+t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c
+k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2)
+t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6:
+(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0
+x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2
+H0)))))) H4))))))))))))) u2 u1 H)))).
+
+theorem pc3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1
+t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t:
+T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda
+(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1)
+(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H
+t2 x H3))))) H1))))))))).
+
+theorem pc3_eta:
+ \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t
+(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead
+(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H:
+(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v
+w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
+(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef
+O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl)
+(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead
+(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t)
+(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S
+O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u)
+H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w
+(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u))))
+(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
+(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl
+c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/props.ma".
+
+include "LambdaDelta-1/pr3/subst1.ma".
+
+theorem pc3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall
+(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d
+c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c
+a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1:
+T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda
+(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x:
+T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T
+(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2
+x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d
+x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x
+(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda
+(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3
+a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0
+(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2
+H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3)))))
+(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/props.ma".
+
+include "LambdaDelta-1/wcpr0/getl.ma".
+
+theorem pc3_wcpr0__pc3_wcpr0_t_aux:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall
+(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3
+(CHead c2 k u) t1 t2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k:
+K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
+(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0:
+T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4
+t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3:
+(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead
+c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4
+t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead
+c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda
+(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2
+k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda
+(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d
+(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6
+t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c
+(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead
+d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2:
+C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
+T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda
+(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7:
+T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6
+t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t
+x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6
+t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x
+H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k
+u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr)
+H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
+
+theorem pc3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1
+t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0:
+C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1:
+T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def
+(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T
+(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k
+u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3
+(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x
+(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2
+(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x
+H6)))))) H4))))))))))))) c1 c2 H))).
+
+theorem pc3_wcpr0:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
+T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind
+T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2
+x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2
+(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/fwd.ma".
+
+include "LambdaDelta-1/subst0/dec.ma".
+
+include "LambdaDelta-1/T/dec.ma".
+
+include "LambdaDelta-1/T/props.ma".
+
+theorem nf0_dec:
+ \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t1 t2))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to
+(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl
+(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T
+(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda
+(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n)
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
+(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl
+(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T
+(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda
+(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n)
+t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T
+(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t:
+T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0
+t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or
+(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2)))
+(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b:
+B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0)
+t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind
+Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2:
+T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda
+(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in
+(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O)
+O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead
+(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t
+t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0
+(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T
+(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S
+O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2)
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0)
+t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind
+Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let
+H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
+\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O
+x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S
+O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6
+P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3)))
+(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x)
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3)
+\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2)
+t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O)
+O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind
+Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t
+(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S
+O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2)))
+H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t
+t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t
+t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
+T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0
+t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0
+t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0
+(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda
+(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
+(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))
+(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0
+t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def
+(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0
+H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3:
+T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3:
+T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead
+(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3))
+(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0
+t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2:
+T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))
+(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead
+(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t
+t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst)
+t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P:
+Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0
+(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind
+Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t
+x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0)
+(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2:
+T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2:
+T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal
+T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3)))
+(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
+t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
+(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))))
+(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P:
+Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead
+(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind
+Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x
+t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
+\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0)
+(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2:
+T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T
+t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P))))))
+(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x
+\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or
+(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
+(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0
+(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift
+(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T
+(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind
+Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let
+H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to
+(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead
+(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2)
+\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2)
+\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead
+(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda
+(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))
+(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t
+t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0
+t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t
+x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def
+(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12
+t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead
+(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda
+(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3:
+T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead
+(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3))
+(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9:
+(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let
+H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3
+(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq
+T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x)
+(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
+(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead
+(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2
+H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T
+t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
+(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P:
+Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0
+(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind
+Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void)
+t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0)
+(THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2:
+T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2:
+T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal
+T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7))
+H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
+t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
+(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P:
+Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0
+(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind
+Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void)
+x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
+\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0)
+(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2:
+T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq
+T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t)
+P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4)))
+(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda
+(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda
+(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2
+t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2:
+T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead
+(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t
+t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void)
+t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S
+O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S
+O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t
+(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead
+(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y
+(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void
+not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f:
+F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0)
+t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def
+H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
+T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P:
+Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq
+T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda
+(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
+u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T
+(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0
+(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T
+(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3:
+T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind
+x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t
+t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq
+T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3:
+T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall
+(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or
+(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to
+(eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2:
+T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2))
+t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2)
+t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0
+(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat
+Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1
+x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead
+(Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1
+(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat
+Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl)
+(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead
+(Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind
+Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7))))
+(pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2
+(pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst)
+x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2:
+T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda
+(t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2:
+T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead
+(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1
+x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead
+(Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2)
+(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead
+(Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat
+Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1
+t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2:
+T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2)
+t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind
+Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2))
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void)
+x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat
+Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T
+(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1
+x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2))
+(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead
+(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P:
+Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow
+(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S
+O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst
+Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl
+x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P:
+Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq
+T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t
+t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
+T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0
+t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0
+t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0
+(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda
+(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
+(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))
+(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl)
+t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead
+(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0
+x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def
+(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def
+(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0
+H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0)
+in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl)
+t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3:
+T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t
+t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq
+T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0
+x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t
+x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda
+(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0
+(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead
+(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3:
+T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b)
+w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in
+(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat
+Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind
+Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t
+(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2
+H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
+(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b)
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat
+Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift
+(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda
+(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2:
+T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))
+(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not
+(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda
+(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
+x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0
+x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
+x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def
+(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3
+t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda
+(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead
+(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10)
+in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat
+Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))))
+(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1
+x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4
+(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11)))))))))))))
+H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2:
+T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0
+t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t
+t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
+T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T
+t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror
+(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat
+Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2)
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0)
+t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))
+(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead
+(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat
+Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x
+(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x
+(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in
+(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat
+Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda
+(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))
+(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead
+(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t
+t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl)
+t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P:
+Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead
+(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat
+Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x
+t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
+\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0)
+(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2:
+T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq
+T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t)
+P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3)))
+H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq
+T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
+Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat
+Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0)
+t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0
+(pr0_refl t0) t))) f)) k)))))) t1).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+inductive pr0: T \to (T \to Prop) \def
+| pr0_refl: \forall (t: T).(pr0 t t)
+| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1:
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1)
+(THead k u2 t2))))))))
+| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to
+(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1
+(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2))))))))
+| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1:
+T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0
+u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead
+(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t2)))))))))))))
+| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1:
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to
+(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
+| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall
+(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O
+t1)) t2))))))
+| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u:
+T).(pr0 (THead (Flat Cast) u t1) t2)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/props.ma".
+
+theorem pr0_gen_sort:
+ \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
+\def
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq
+T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0:
+T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T
+t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n))
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u:
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_:
+(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1))
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
+(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda
+(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_:
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1))
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1
+(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead
+(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1)
+(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x
+H0))) H))).
+
+theorem pr0_gen_lref:
+ \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
+\def
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq
+T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0:
+T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T
+t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n))
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u:
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_:
+(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1))
+(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
+(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda
+(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_:
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1))
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1
+(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead
+(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1)
+(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x
+H0))) H))).
+
+theorem pr0_gen_abst:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1
+t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y:
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T
+t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda
+(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def
+(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r
+T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind
+Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1
+(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda
+(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0)
+(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind
+Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind
+Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind
+Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind
+Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead
+(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T
+u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10))))
+H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl)
+v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind
+b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2
+w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1
+t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind
+Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
+Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O
+t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in
+((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
+t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
+(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow
+(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda
+(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t))
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T
+(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12
+\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_:
+False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_:
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u
+t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
+t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))).
+
+theorem pr0_gen_appl:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1
+t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b)
+v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2))))))))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y:
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T
+t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
+t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t:
+T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T
+T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead
+(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
+t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl
+t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0
+u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind
+Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq
+T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0
+u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda
+(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2
+t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq
+T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let
+H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def
+(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind
+T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda
+(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
+(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2
+(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7))
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl)
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
+T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat
+Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u
+t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat
+Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1
+(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1
+(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12
+\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in
+(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda
+(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl)
+(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead
+(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind
+Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2)
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0
+y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2
+(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2
+t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not
+(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1
+v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda
+(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl)
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat
+Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat
+Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0)
+| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow
+t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1)
+H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t:
+T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1
+(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def
+(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in
+(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1
+t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in
+(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat
+Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat
+Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat
+Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
+Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl)
+(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0))
+(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)))
+H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2:
+T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_:
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead
+(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2
+w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not
+(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead
+(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def
+(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq
+T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T
+(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
+(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
+True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
+
+theorem pr0_gen_cast:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))
+(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda
+(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat
+Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat
+Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0:
+T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat
+Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
+u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T
+(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda
+(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda
+(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0)
+(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat
+Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat
+Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat
+Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat
+Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0
+(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0
+(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda
+(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0
+t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2
+t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind
+Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2
+t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda
+(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead
+(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
+\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
+True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in
+(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq
+T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T
+(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b:
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u:
+T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat
+Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
+(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0)
+(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
+u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))).
+
+theorem pr0_gen_abbr:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda
+(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t:
+T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T
+T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead
+(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0
+t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind
+Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda
+(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0
+t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind
+Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return
+(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
+(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
+H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0
+| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
+H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
+H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind
+Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let
+H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to
+(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in
+(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2
+u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in
+(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind
+Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
+t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind
+Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7))
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_:
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O
+v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
+(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S
+O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_:
+(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead
+(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
+T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S
+O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq
+T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1
+t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind
+(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0
+(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0
+t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3:
+(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq
+T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in
+(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
+t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t:
+T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
+t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or
+(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O
+u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro
+T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind
+Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1
+y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7)))))))))))))
+(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda
+(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind
+Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S
+O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S
+O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in
+((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
+t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
+(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow
+(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda
+(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t))
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3
+(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))
+(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0:
+T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3)))))))
+(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_:
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
+(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))
+H4)))))))) y x H0))) H)))).
+
+theorem pr0_gen_void:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O)
+O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda
+(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda
+(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void)
+u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
+t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
+(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1
+t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2:
+T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda
+(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal
+T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0
+u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda
+(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2
+t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t:
+T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T
+t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0
+t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead
+(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void)
+u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead
+(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B
+b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl)
+v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T
+(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)))))
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda
+(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T
+(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w)
+(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead
+(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B
+b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
+t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda
+(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1
+t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k
+_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O
+t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
+(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T
+T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t))
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O)
+O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift
+(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2
+H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
+(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x
+H0))) H)))).
+
+theorem pr0_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0
+(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(pr0 t1 t2)))))))
+\def
+ \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t
+x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat
+d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T
+x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t:
+T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq
+T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t:
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1
+x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2:
+T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq
+T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0))))))
+(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2:
+((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T
+(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0
+t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2
+t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
+x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
+T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda
+(k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4:
+T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))
+(\lambda (b: B).(\lambda (H6: (eq T (THead (Bind b) u1 t2) (lift h x1
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
+b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
+(t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
+(Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T
+t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h
+(S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T
+(THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b)
+x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1)
+x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) x4) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T (\lambda (t4:
+T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda
+(t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda
+(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T
+(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b)
+t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b)
+x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1
+x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind
+b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h x1 (THead (Bind b) x5
+x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift_bind b x5 x4 h
+x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 H_x0)))) (H2 x2 x1 H8)) t3
+H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind b u1 t2 x0 h x1 H6))))
+(\lambda (f: F).(\lambda (H6: (eq T (THead (Flat f) u1 t2) (lift h x1
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+f) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda
+(t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
+(Flat f) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T
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+(\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4:
+T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
+(\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f)
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+(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H10: (pr0
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+(Flat f) x2 x3) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4)))
+(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f)
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+(t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq
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+x4)) (lift_flat f x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2
+H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat
+f u1 t2 x0 h x1 H6)))) k H5))))))))))))) (\lambda (u: T).(\lambda (v1:
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+T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
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+(Bind Abst) u t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda
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+h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0:
+T).(\lambda (z: T).(eq T x3 (THead (Bind Abst) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind
+Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3)
+t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H9: (eq T x3 (THead (Bind
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+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda
+(t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T
+(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda
+(x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H12: (pr0 x5
+x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(eq T (THead (Bind Abbr) v2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex2_ind T (\lambda
+(t4: T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T
+(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 (lift h (S x1) x6)) (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5))
+t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T v2 (lift h x1 x7))).(\lambda
+(H13: (pr0 x2 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda
+(t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6)) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))))
+(ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h
+(S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
+(THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h
+x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S
+x1) x6)) (lift_bind Abbr x7 x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2
+H_x0)))) (H2 x2 x1 H7)) t3 H_x)))) (H4 x5 (S x1) H11)) x3 H9))))))
+(lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat Appl v1
+(THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b: B).(\lambda
+(H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
+v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h
+x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2:
+T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
+u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1
+x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2:
+T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2
+t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
+x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
+T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T
+(THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T
+(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
+(\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (THead (Bind b) u1 t2) (lift h x1 z)))) (ex2 T
+(\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl) x2
+x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead
+(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
+(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0
+z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda
+(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1)
+x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T
+(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4))
+(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
+(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift
+h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
+(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq
+T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S
+x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead
+(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1
+x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift
+(S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4:
+T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O
+v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
+Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T
+v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7)
+(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))))
+(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T
+(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t
+(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
+Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat
+Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
+(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat
+Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
+(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat
+Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead
+(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S
+x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat
+Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5
+x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1)
+x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1
+x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2
+H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12))))))
+(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1
+(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0:
+T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
+T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w:
+T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
+Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
+(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
+(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9:
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda
+(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1
+t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3
+(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3
+(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3
+(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0:
+(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1
+x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w)
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let
+H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11
+(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in
+(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5)
+(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq
+T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6:
+T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17:
+(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T
+(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O
+x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3)
+t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5
+x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6))
+(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17))
+w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1
+H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1
+H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u:
+T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u
+(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda
+(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
+T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2)
+(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
+(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1
+x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T
+(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift
+h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S
+O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1))
+(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1)
+(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O))
+H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4:
+T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4:
+T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift
+h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t)
+t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
+T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5:
+T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4
+x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
+t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O
+x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5
+(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4
+x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0
+H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u:
+T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast)
+u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T
+x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift
+h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast)
+x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h
+x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
+(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0
+x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
+T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T
+t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4:
+T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
+Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3
+H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1
+H3)))))))))) y x H0))))) H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/fwd.ma".
+
+include "LambdaDelta-1/lift/tlt.ma".
+
+theorem pr0_confluence__pr0_cong_upsilon_refl:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3:
+T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to
+(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x)
+\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4))
+t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t)))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda
+(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda
+(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda
+(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0
+(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4
+t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5)
+(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S
+O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind
+b))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_upsilon_cong:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2:
+T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall
+(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5:
+T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t))
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t)))))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda
+(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2
+x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2
+x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda
+(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t))
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0))
+(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat
+Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp
+(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat
+Appl)) (Bind b))))))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_upsilon_delta:
+ (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w:
+T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x:
+T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2
+x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to
+((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t))))))))))))))))))))
+\def
+ \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2:
+T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2:
+T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2:
+(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2
+x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda
+(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T
+(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0
+(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0
+(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead
+(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H
+u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O
+v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2)
+(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind
+Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2:
+T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda
+(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl)
+u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3
+(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8:
+(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))
+(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon
+Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift
+(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O)
+O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl))
+(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9
+(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1
+H5))))))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_upsilon_zeta:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3:
+T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0
+u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1:
+T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat
+Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
+(lift (S O) O v2) (lift (S O) O x))) t)))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda
+(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda
+(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x:
+T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4:
+(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O
+(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1
+t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat
+Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl)
+(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O)
+O)))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_delta:
+ \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to
+(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall
+(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind
+Abbr) u3 w) t))))))))))))))
+\def
+ \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3
+t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda
+(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3
+x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2:
+T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))
+(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3
+x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2))
+(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w
+w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)))
+(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0
+x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta
+u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4))
+(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).
+
+theorem pr0_confluence__pr0_upsilon_upsilon:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2:
+T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1:
+T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to
+(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2
+x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl)
+(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) t)))))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
+(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2
+x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1
+x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1)
+t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0)
+x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat
+Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0)
+(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1
+H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O
+x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S
+O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).
+
+theorem pr0_confluence__pr0_delta_delta:
+ \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to
+(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to
+(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0)
+\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))))))))))))))))
+\def
+ \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2
+t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0:
+(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2:
+(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5
+x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2:
+T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0
+x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2:
+T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w
+x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp
+u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr))))
+(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O
+x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0
+O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7:
+(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr))
+(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x
+H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2:
+T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda
+(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2
+w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1:
+T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0
+w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0
+w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1
+H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda
+(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T
+(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda
+(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t:
+T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1)
+(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2
+x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in
+(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x
+H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda
+(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x
+x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t:
+T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3:
+T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1
+x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta
+u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11))
+(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))
+(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2
+w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9
+(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0
+x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5))
+(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
+
+theorem pr0_confluence__pr0_delta_tau:
+ \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to
+(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2
+t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2
+t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda
+(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda
+(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
+(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S
+O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t:
+T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w
+(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
+(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))))))))
+(pr0_gen_lift t4 t3 (S O) O H0)))))))).
+
+theorem pr0_confluence:
+ \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0
+t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))
+\def
+ \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to
+(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3))
+(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall
+(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0
+v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2
+t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2:
+T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4
+t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2
+t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3
+t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1)
+\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))))
+(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5:
+T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0
+return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4
+t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6:
+T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4
+t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2)
+\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))
+(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6:
+T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t
+(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda
+(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5:
+T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5:
+T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall
+(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8:
+T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5:
+T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall
+(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8:
+T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T
+(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def
+(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T
+(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1)
+(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5)
+H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead
+k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1
+t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4
+t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
+t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5)
+(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1
+u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6:
+T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda
+(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k
+u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2
+(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall
+(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8:
+T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0
+t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0
+(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2
+t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12))))
+t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow
+(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4))
+t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr)
+v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0
+t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind
+Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0
+v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
+(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4
+t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead
+(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl)
+v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6
+t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def
+(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead
+(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6:
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
+T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7)
+in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind
+Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead
+(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind
+Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1
+v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl)
+v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
+v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1
+t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B
+b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda
+(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1
+u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6:
+T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2
+T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t
+(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1
+t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v:
+T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v
+t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8
+t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in
+(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13
+(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta
+u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1
+t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead
+(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to
+((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda
+(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda
+(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5
+w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead
+(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6:
+T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6))
+H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6:
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
+T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T
+(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0
+(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11
+t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10))
+t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7:
+(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5
+t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5
+t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5
+t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to
+(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))
+(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def
+(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O)
+O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6:
+T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let
+H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u
+(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6:
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
+T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in
+(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6))
+(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl
+t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5
+H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda
+(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T
+t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
+(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda
+(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t
+(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T
+(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6
+t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda
+(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def
+(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7:
+T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
+T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u
+t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6))
+(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1
+H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t)
+(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) |
+(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1
+t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3)
+(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4)
+\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))
+(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda
+(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5
+t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5:
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0
+t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5
+t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0
+t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T
+(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))))
+(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1
+t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7:
+T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def
+(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let
+H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to
+(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T
+(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead
+k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6))
+(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k
+u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5
+(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow
+(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3
+t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6)
+t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead
+k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda
+(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5
+t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7))
+H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0
+t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1
+| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in
+((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _
+t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20:
+(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r
+K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7))
+(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1
+(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3
+(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0
+u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x:
+T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda
+(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7:
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))
+(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3
+x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7:
+T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25
+k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2
+H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21)))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10)
+\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind
+Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
+(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2
+t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14:
+(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda
+(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind
+Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let
+H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat
+Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in
+(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead
+k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let
+H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24
+\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5)
+H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8:
+T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T
+t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl
+t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda
+(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq
+T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T
+(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda
+(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda
+(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8:
+T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0
+(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0
+(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2
+x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead
+(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6
+H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1
+(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4
+H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3
+H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead
+(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9)
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _)
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K
+(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead
+k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind
+Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda
+(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9
+u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2
+t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))))
+(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead
+(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda
+(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead
+(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8)
+t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to
+((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9)
+t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_:
+(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8
+t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat
+Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind
+Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38:
+(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2
+t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0:
+T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x)
+(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind
+Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2
+H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat
+Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5)
+(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4
+H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0
+(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8
+H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind
+Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
+\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5)
+H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26)))
+| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda
+(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst)
+u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S
+O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead
+(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _
+_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5)
+H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26
+H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28:
+(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq
+T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr)
+u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
+\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T
+(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O
+u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26
+H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T
+(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda
+(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10)
+\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in
+lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match
+t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0
+d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
+t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind
+Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S
+O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B
+(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match
+k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u
+t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S
+O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
+T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0
+u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8
+t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10:
+T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind
+Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind
+T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst))
+\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4)
+t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda
+(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to
+((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4)
+t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda
+(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match
+(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
+(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead
+(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5
+H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25
+H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat
+Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let
+H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in
+(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21)))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0
+u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1
+(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
+v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7:
+T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6)
+\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0
+t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1
+v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat
+Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda
+(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
+| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
+(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let
+H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0
+t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat
+Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in
+(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead
+k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
+(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7:
+T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7:
+T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in
+pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T
+t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0
+(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7)
+\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30:
+(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4)
+\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))
+(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0
+t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8)
+t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8:
+T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat
+Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2
+x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16
+u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1
+(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind
+b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow
+(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32:
+(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
+(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7)
+(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 |
+(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7)
+(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e:
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
+(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7)
+(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4
+u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9:
+T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4)
+t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S
+O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda
+(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8)
+\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
+(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
+t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead
+(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T
+(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10))))))
+(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda
+(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))
+(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6
+x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9))
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8))
+t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44:
+(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0
+v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
+(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda
+(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45
+H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1
+(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b)
+u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b)
+u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3
+H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1
+(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl)
+v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5
+H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33))
+H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31:
+(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0
+t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def
+(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
+K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T
+(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33))
+H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32)
+\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4
+t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead
+(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat
+Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0)
+u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst))
+\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b)
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31
+H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq
+T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T
+(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
+(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind
+Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9]))
+(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _)
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7)
+(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0)
+\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5)
+\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0
+(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T
+u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind
+Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to
+(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10:
+T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
+t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq
+T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0
+O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10))
+(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O
+v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w)
+t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to
+((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead
+(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0
+u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43
+\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat
+Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to
+(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11))
+(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def
+(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr
+H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst)))
+H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9:
+T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0
+t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9))
+(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl)
+u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda
+(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9:
+T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0
+(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0
+(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))
+(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2
+x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51
+t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead
+(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr)
+u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind
+Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40
+u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat
+Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5)
+(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19))))))))
+t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35))
+H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda
+(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0
+t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let
+rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9
+with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
+(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0
+u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))])
+in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match
+t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0
+d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
+t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b)
+u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S
+O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda
+(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0
+| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in
+(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5)
+\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))
+(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O
+t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2
+T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
+t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O)
+O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8)
+\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
+(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
+t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not
+(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead
+(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b
+Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda
+(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0
+t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11)
+\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11
+t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5
+(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37)
+in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O)
+O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9)))
+(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl)
+u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
+(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O
+x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9:
+T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
+(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9))
+t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9))
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
+x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9))
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
+x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0
+v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48
+H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b)
+u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b)
+u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O)
+O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1
+(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44))))
+(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u
+(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) |
+(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u
+t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def
+(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0
+t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5))
+(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10
+H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12:
+(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr)
+u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead
+(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6
+w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8:
+T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w)
+t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to
+((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0
+u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
+(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind
+Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
+(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead
+(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K
+k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr)
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1
+(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3
+(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))
+(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7))
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0
+u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w
+H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2
+H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k
+H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5
+t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O
+t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O)
+O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5
+t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8:
+T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7:
+T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0
+(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not
+(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t
+(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O)
+O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind
+b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3)
+(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7]))
+(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20:
+(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O
+t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7:
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def
+(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind
+T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T
+(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7))
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O
+x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7:
+T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8:
+T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7:
+T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O
+x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0
+x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead
+(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta
+b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26
+t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18))
+H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u)
+\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11:
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8))
+(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2
+(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2
+t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let
+H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
+(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat
+Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
+(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead
+(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k
+(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast)
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda
+(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7
+u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8
+t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda
+(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda
+(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20
+t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15))))
+t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t)
+(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4
+H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst)
+u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead
+(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda
+(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T
+(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda
+(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5
+t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5:
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5
+t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_:
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2))
+H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7))))
+(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat
+Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7)
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9))
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind
+Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2
+t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4))
+(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq
+T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda
+(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6)
+t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind
+Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T
+(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2)
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4)
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda
+(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let
+H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) |
+(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow
+(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq
+T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda
+(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda
+(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda
+(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def
+(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3)
+H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8:
+T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T
+t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl
+t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda
+(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq
+T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T
+(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda
+(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda
+(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5
+(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u
+t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v:
+T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v
+t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10
+t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let
+H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T
+(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead
+(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30:
+(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0
+(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2
+(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4
+t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1
+(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6
+H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3
+H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead
+(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9)
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _)
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K
+(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead
+k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
+T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat
+Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda
+(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9
+u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2
+t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))))
+(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead
+(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda
+(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead
+(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8)
+t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to
+((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_:
+(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda
+(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19)
+in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9)
+\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to
+(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11
+t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let
+H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T
+(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda
+(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead
+(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda
+(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0
+t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead
+(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8
+x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead
+(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0
+H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans
+(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3))
+(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst)
+u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind
+Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0
+(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24
+H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T
+(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u
+t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def
+(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
+K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T
+(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
+(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24
+v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat
+Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29:
+(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let
+H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
+K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not
+(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
+(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) |
+(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead
+(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead
+(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7)
+(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow
+(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match
+b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
+\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
+False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind
+Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) |
+(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind
+b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8
+t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10)
+\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in
+lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match
+t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0
+d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
+t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind
+Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S
+O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B
+(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match
+k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u
+t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S
+O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0
+u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8
+t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10:
+T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat
+Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind
+T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst))
+\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda
+(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to
+((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda
+(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match
+(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
+(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead
+(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3
+H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24
+H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat
+Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let
+H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in
+(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6
+H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind
+T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T
+(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2
+t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T
+(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
+T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5
+t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
+(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0
+t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
+| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])]))
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead
+(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
+| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst)
+u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_:
+(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7
+v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4))
+H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0
+t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25:
+(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2
+t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))
+(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3
+x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x)
+(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind
+Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2
+H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat
+Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5)
+(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15))))))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1
+u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0
+(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0
+(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0
+v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15:
+(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6))
+t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6))
+(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8))
+(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b
+Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
+t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl)
+v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1
+t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
+| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
+(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])]))
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead
+(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead
+_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3))
+(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u
+u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst
+(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
+(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O
+v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
+Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in
+False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat
+Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22))
+H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6
+H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5)
+t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
+Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
+(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda
+(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w)
+(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
+T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5
+t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda
+(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead
+(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)))
+H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0)
+\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5))
+t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O
+t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6)
+\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
+T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7:
+T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0
+(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_:
+(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t
+(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7))
+H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2
+t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6
+H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5)
+t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
+(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq
+T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))
+(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0
+t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst)
+u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
+(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T
+t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2)))))
+t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5)
+\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to
+((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
+t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2)
+\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6))
+(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b
+Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda
+(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5:
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5)
+\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind
+T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7:
+T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_:
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r
+T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b)
+u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17
+\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6:
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
+T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6)
+in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)))
+(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10
+(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13)
+H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T
+(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T
+(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3)
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
+(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda
+(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
+T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5
+t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) |
+(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1
+t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat
+Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead
+(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1
+u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0:
+K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat
+Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19
+(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return
+(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead
+(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow
+(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7
+t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2
+T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))
+(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1
+t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat
+Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T
+(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32
+\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall
+(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda
+(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead
+(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1
+(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0
+v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
+T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x:
+T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0
+(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b
+H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl)
+u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead
+(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow
+(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31:
+(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
+(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7)
+(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 |
+(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7)
+(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e:
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
+(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7)
+(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4
+u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3
+t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9:
+T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat
+Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda
+(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8)
+\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
+t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead
+(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T
+(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10))))))
+(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def
+(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26
+(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9:
+T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to
+(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12))
+(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind
+b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2))
+H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0
+u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3
+(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2
+x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9))
+(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead
+(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45:
+(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5
+t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0
+(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda
+(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat
+Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x
+H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead
+(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx
+(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38
+u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl)
+u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat
+Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx
+(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7
+(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b)
+H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29)
+\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u
+t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8)
+t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7))
+(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow
+(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False
+| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind
+((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))
+H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31)
+\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4
+t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead
+(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat
+Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0)
+u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst))
+\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29
+H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31:
+(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T
+(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
+(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind
+Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9]))
+(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _)
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7)
+(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1)
+\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5)
+\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda
+(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4
+u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind
+Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to
+(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
+t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq
+T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0
+O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat
+Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w)
+t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to
+((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead
+(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10:
+T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1
+u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42
+\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23
+Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
+Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr
+H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat
+Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def
+(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10:
+T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12:
+T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat
+Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1
+(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0
+v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead
+(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x:
+T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda
+(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9:
+T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9)))
+(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9))
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead
+(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5
+x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead
+(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O)
+O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x
+H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind
+Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx
+(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5
+H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat
+Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3)
+(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12)))))
+(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3
+H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b
+H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u)
+\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead
+(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T
+\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0
+d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T
+\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0
+d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
+t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b)
+u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S
+O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda
+(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0
+| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in
+(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3)
+\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T
+(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))
+(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O
+t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2
+T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10))))))))
+(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7)
+(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to
+(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
+t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not
+(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10:
+T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b
+Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda
+(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let
+H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t))
+H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t
+(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v
+t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10
+t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead
+(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda
+(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9:
+T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda
+(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x:
+T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7
+x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10))
+(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def
+(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda
+(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
+x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0:
+T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T
+(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda
+(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S
+O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda
+(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T
+(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta
+b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans
+(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind
+b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx
+(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42
+u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46
+u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8
+t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33))
+H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T
+(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8
+t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match
+e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8
+t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead
+(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead
+(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t
+H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda
+(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda
+(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0
+(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6)
+t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
+T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6)
+t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda
+(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t
+(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6
+(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _)
+\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat
+Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B
+(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match
+t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind
+Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead
+_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow
+t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0
+(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26:
+(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
+Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2))
+H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b
+(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match
+(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
+(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31))
+b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) |
+(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17:
+(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T
+(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T
+(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead
+(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0
+Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind
+b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind
+b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B
+b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0
+Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda
+(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
+(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead
+(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24)
+in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _
+_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
+\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead
+(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match
+t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24)
+in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead
+_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow
+t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0
+(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30:
+(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
+b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2))
+H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0
+(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6))
+t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0
+H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29)
+in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in
+(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T
+(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl)
+(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4
+x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7))
+(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0
+(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda
+(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T
+(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda
+(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O)
+O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42:
+(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2
+u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead
+(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0
+t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0)
+u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3
+H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0
+(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat
+Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27))
+H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13
+t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5)
+t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind
+Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0
+u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda
+(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w)
+t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to
+((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
+T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5
+t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda
+(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead
+(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) |
+(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind
+b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead
+(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B
+b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2
+t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not
+(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
+T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5
+t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in
+(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20)
+in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6
+(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u)
+\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15:
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2
+t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5
+t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_:
+(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
+Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in
+(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5)
+H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))
+H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t)
+(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2
+t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3)
+t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind
+Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1
+u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0
+t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind
+Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1
+u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0
+t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1
+u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11
+\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_:
+(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
+[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T
+t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))
+(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let
+H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr)
+u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6
+t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead
+(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6:
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
+T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T
+(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0
+(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead
+(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq
+T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12
+k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T
+(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead
+k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
+(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda
+(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0
+u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
+T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) |
+(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
+H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1
+| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
+H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
+| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
+H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr)
+k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))))
+(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13
+(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1
+(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3
+(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7)))
+(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7))
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0
+u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3
+t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w
+H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5)
+u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6
+H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta
+u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1
+(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6)
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_:
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8:
+T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6)
+t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
+(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0
+t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr)
+u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let
+H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in
+(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13
+H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14)
+\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0
+t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S
+O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5))
+(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7
+t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead
+(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b)
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15
+H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow
+(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T
+(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda
+(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6)
+\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr)
+u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead
+(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7:
+T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))
+(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19:
+(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _)
+\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in
+((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _
+t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5)
+H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda
+(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to
+(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in
+(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let
+H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T
+(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28:
+(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0
+u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w
+H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0
+t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6
+H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5
+t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O
+t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O)
+O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5
+t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda
+(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda
+(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))
+(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead
+(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr |
+(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18)
+in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead
+_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift
+(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3)
+(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1
+u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0:
+B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b
+(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr
+H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def
+(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind
+T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T
+(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7))
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
+T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O
+x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7:
+T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7:
+T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0:
+T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2
+x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl
+(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30
+t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6
+(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u)
+\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13:
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2
+w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))
+(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17
+\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17))))
+t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t)
+(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3
+u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3))
+t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O
+t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4)
+\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))
+(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b
+Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda
+(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8:
+(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda
+(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
+[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5
+t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t
+t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7))
+(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6:
+T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T
+(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t
+(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in
+(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to
+(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T
+(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead
+(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1
+t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1
+(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11)))
+t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow
+(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6)
+t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6)
+t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6)
+\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))
+(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r
+T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4
+(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
+_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
+(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7]))
+(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7:
+T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d
+u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
+(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort
+n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map
+f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O
+t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K
+(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
+t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r
+K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23
+\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19)
+in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7:
+T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
+(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S
+O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda
+(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let
+H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
+(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
+(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x)
+(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
+(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7))
+(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
+(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0:
+T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift
+(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx
+(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O)
+O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta
+u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1
+(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6)
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_:
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
+(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind
+Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_:
+(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda
+(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl)
+v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b)
+u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))
+H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2
+H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead
+(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat
+Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead
+(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat
+Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2)
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not
+(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
+(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0)
+u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1
+t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7)))
+H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6
+H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5)
+t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
+Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
+t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind
+Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
+t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda
+(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r
+T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4
+(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in
+((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7
+_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr)
+u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in
+lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
+t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7)
+\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1
+t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24
+\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21)
+in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7:
+T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift
+(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda
+(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in
+(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
+(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
+(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6
+(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in
+(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w)
+t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1
+x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1)
+(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S
+O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1
+H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t
+H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda
+(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T
+t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq
+T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6
+t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))
+(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3))
+t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal
+T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
+\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0
+(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S
+O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7:
+T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1)
+(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O
+t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat)
+(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) |
+(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f
+d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O
+t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u
+u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7:
+T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall
+(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
+T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in
+(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3
+t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1
+Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
+t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))
+x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2
+H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) |
+(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0
+t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5)
+(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5
+t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u
+(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def
+(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6
+t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4
+(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow
+(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4
+t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to
+((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
+t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0
+t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2
+t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return
+(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to
+((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
+t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5
+t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))
+(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t
+(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T
+(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1
+t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda
+(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def
+(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7:
+T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
+T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u
+t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
+(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2
+H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7
+t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda
+(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_:
+T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
+(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7:
+T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13:
+(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
+Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _)
+\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _)
+\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
+\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda
+(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast)
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
+(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T
+(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in
+(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in
+(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6)
+t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
+(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx
+(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9
+H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq
+T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T
+(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind
+Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0
+v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
+(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6)
+t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
+t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead
+(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T
+(THead (Flat Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
+(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
+True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H14) in
+(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b
+H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat
+Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl)
+v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
+v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
+t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B
+b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda
+(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1
+u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
+T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind
+b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1
+(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1
+t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta
+u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr)
+u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T
+(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2)
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_:
+(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let
+H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7))
+H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat
+Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
+t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8
+H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead
+(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T
+(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not
+(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2
+(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq
+B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda
+(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O)
+O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in
+(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
+t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6
+H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda
+(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T
+t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
+(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda
+(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
+T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t
+(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0
+t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast)
+u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3)
+(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8:
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10:
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast)
+u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5
+H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
+t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))
+(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))
+x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11))))))
+H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t)
+(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2
+(refl_equal T t) (refl_equal T t1))))))))) t0).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/defs.ma".
+
+include "LambdaDelta-1/subst0/subst0.ma".
+
+theorem pr0_lift:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall
+(d: nat).(pr0 (lift h d t1) (lift h d t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t)
+(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
+(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0
+(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0
+(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda
+(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t:
+T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2)
+(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k
+d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d)
+t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4))
+(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h
+d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_:
+(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h
+d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
+t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3)
+(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead
+(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u
+t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r
+T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s
+(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t)
+(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h
+d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl)
+(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s
+(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d)
+u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl)
+d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d
+(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s
+(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s
+(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3)))
+(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda
+(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d:
+nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d:
+nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d:
+nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d:
+nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d)
+(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b)
+(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3))
+(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead
+(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O)
+O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead
+(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d))
+t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O
+v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead
+(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift
+h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t)))
+(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h
+d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d
+u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4)))))
+(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat
+Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d)
+t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O)
+d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d
+u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d)
+t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d
+v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b)
+d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S
+O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3))
+(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
+v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3)
+h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
+u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1)
+(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
+t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3)
+(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda
+(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift
+h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr)
+u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr)
+d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind
+Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S
+d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in
+(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2)
+(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d
+(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal
+nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w))
+(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3))
+(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0:
+(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
+t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3)
+(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d:
+nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S
+O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d)
+(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O
+t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t:
+T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift
+h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S
+O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d)))
+(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift
+(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_:
+(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h
+d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d:
+nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d)
+t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast)
+d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u
+t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))).
+
+theorem pr0_subst0_back:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t:
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3)))))))))
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1
+v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t:
+T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0)
+(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1:
+((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t))
+(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
+(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0
+u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x:
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0
+(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3
+H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v:
+T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0
+(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T
+(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t
+t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t
+(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4
+x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
+(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x)
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4:
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t:
+T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4:
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda
+(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t
+(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3
+x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t))
+(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1
+t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3
+t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3
+H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
+
+theorem pr0_subst0_fwd:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t:
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4)))))))))
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v
+u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t:
+T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0)
+(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1:
+((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t))
+(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
+(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0
+u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x:
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0
+(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3
+x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda
+(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_:
+(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to
+(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3
+t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0
+(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4
+x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
+(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x)
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4:
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t:
+T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4:
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda
+(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead
+k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3
+x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t))
+(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1
+t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4)
+t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8
+t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
+
+theorem pr0_subst0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall
+(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1
+v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t2 w2))))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0
+w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i:
+nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1
+v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd
+v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0:
+(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2
+w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
+t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i:
+nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2:
+T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5:
+T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4))
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k
+u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3
+t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq
+T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1
+(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1
+(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3)
+(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2)
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))
+(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead
+k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda
+(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T
+(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2
+(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0
+(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3)
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0:
+T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror
+(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x
+t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T
+(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2
+(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k)
+(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7))))
+H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5)))
+(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq
+T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0
+w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1
+(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k
+u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind
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+x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12:
+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T
+x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst)
+x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1))
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4)
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
+(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead
+(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2))
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14:
+(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1))
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1
+(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
+Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T
+(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s
+(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2))
+(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or
+(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))
+(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind
+Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1
+(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2)
+(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16
+v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1
+H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i)
+H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0
+v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead
+(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat
+Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2
+t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1
+x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3)
+x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3)))
+(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5:
+T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind
+Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T
+(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2:
+T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1
+(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u
+u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda
+(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_:
+(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t:
+T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t:
+T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2))
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0
+x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
+w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
+Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4)
+w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0
+(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4
+H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2:
+T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda
+(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind
+Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead
+(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0
+x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl)
+x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4)
+(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind
+Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10:
+(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5:
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda
+(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind
+Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4))
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead
+(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead
+(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i))
+v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat
+Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat
+Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i
+v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2:
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3
+t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
+Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4)
+w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2:
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat
+Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda
+(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda
+(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
+t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0
+(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4))
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x))
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda
+(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2
+x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
+t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14)
+(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5)))
+(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2:
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3
+t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
+Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4)
+w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s
+(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead
+(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2))
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+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0
+(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
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+(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
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+(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0
+(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4))
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x))
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+(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2
+x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
+t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15)
+(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8
+v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1
+H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
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+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2)))
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0
+t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0
+w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst)
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+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T
+x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst)
+x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3))
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
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+w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda
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+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2))
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+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2:
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+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
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+t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15)
+(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5)))
+(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2:
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+u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i))
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+(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x
+(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0
+x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1))
+(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O)
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+w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1
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+v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat
+Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl)
+i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T
+x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat
+Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b)
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))
+(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)))
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x:
+T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s
+(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t:
+T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or
+(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T
+(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat
+Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x))
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda
+(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2:
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat
+Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
+(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0
+(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))
+(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2
+x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
+b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd
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+(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind
+b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl))
+u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2:
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s
+(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead
+(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)))
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x))
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x
+x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind
+(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i
+v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0
+(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b)
+u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead
+(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2
+H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O)
+O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl)
+v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3
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+(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))
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+x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
+b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd
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+(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4
+x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b)
+(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T
+(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead
+(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i)
+v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat
+Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind
+b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda
+(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind
+T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b)
+x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3))
+(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O)
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind
+(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s
+(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead
+(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2
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+i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead
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+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2:
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+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead
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+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
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+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind
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+(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i
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+(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2
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+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
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+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift
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+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2))
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+(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H22 x2 x H20 x3 t4 H18) (subst0_fst
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+w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2:
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11
+w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2:
+T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda
+(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x:
+T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T
+(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2
+w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0
+O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def
+(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in
+(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w
+x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x
+H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18))))))))
+(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2
+H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2:
+T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1
+w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind
+Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13:
+(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2:
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind
+Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead
+(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2
+w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2
+x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind
+Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x
+x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0
+(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead
+(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr)
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))
+(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd
+(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind
+Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14:
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2
+u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0
+x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O
+x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind
+Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4
+x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal
+nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20
+\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S
+i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t))
+(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead
+(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1)
+w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda
+(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22:
+(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2
+H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4
+(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21)))))))
+(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S
+i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i
+H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7))
+(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1:
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2)
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda
+(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift
+(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda
+(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b)
+u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or
+(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b)
+u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda
+(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6:
+(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u
+x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda
+(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5:
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda
+(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b)
+i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6:
+(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift
+(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5)))
+(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda
+(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T
+x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in
+(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t
+t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n
+v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2:
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind
+b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u
+(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda
+(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4)
+(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2))
+(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda
+(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2
+t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda
+(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0
+x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u
+(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift
+(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T
+(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1
+x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S
+O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i)
+v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O
+t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or
+(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O
+x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11
+\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift
+(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda
+(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda
+(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or
+(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0
+(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S
+O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O
+x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0)))
+(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0
+i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x
+x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0
+(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift
+(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda
+(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1
+x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S
+O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u
+(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1:
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2)
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda
+(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3)
+w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda
+(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u
+u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda
+(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4:
+(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2:
+T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat
+Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
+(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda
+(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t:
+T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2
+T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4:
+(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T
+w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1
+t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat
+Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T
+(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2:
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4)
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast)
+i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
+(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0
+x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4
+w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead
+(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
+T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4
+x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0
+(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
+(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat
+Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2:
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast)
+i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
+(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0
+x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda
+(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0
+(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast)
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0)))
+(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0
+(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2))
+(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat
+Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1
+x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead
+(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1)
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2:
+T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))
+x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3))
+w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1
+t2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/props.ma".
+
+include "LambdaDelta-1/subst1/defs.ma".
+
+theorem pr0_delta1:
+ \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall
+(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead
+(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1:
+(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind
+Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind
+Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H
+t1 t2 H0 t0 H2))) w H1)))))))).
+
+theorem pr0_subst1_back:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1:
+T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda
+(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x
+H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
+
+theorem pr0_subst1_fwd:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1:
+T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda
+(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x
+H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
+
+theorem pr0_subst1:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall
+(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1
+v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2
+w2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1:
+T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1
+w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to
+(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))))))
+(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0
+t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2))))
+(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda
+(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2))
+(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2))
+(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2
+T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3
+(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2))
+(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0
+w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0
+w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4:
+(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2:
+T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i
+v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/defs.ma".
+
+inductive pr1: T \to (T \to Prop) \def
+| pr1_refl: \forall (t: T).(pr1 t t)
+| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3:
+T).((pr1 t2 t3) \to (pr1 t1 t3))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr1/props.ma".
+
+include "LambdaDelta-1/pr0/pr0.ma".
+
+theorem pr1_strip:
+ \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0
+t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
+(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4:
+T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
+(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
+(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda
+(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda
+(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T
+(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5:
+T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda
+(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
+t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let
+H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t:
+T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)))
+(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
+H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3
+t2 H0)))))))))) t0 t1 H))).
+
+theorem pr1_confluence:
+ \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0
+t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
+(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4:
+T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
+(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
+(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2:
+T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_:
+(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda
+(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda
+(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def
+H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t))
+(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x:
+T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x
+H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda
+(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
+t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
+H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr1/defs.ma".
+
+include "LambdaDelta-1/pr0/subst1.ma".
+
+include "LambdaDelta-1/subst1/props.ma".
+
+include "LambdaDelta-1/T/props.ma".
+
+theorem pr1_pr0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H
+t2 (pr1_refl t2)))).
+
+theorem pr1_t:
+ \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2
+t3) \to (pr1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3)))))
+(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda
+(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0
+t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2
+t5 H3)))))))))) t1 t2 H))).
+
+theorem pr1_head_1:
+ \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall
+(k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t:
+T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k
+t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda
+(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda
+(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing
+(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k
+t3 t) H2))))))) u1 u2 H))))).
+
+theorem pr1_head_2:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall
+(k: K).(pr1 (THead k u t1) (THead k u t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u:
+T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u
+t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing
+(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k
+u t4) H2))))))) t1 t2 H))))).
+
+theorem pr1_comp:
+ \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u:
+T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u))))))))
+\def
+ \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t:
+T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall
+(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda
+(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2
+t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1
+t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t:
+T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead
+k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t
+u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k
+t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1
+H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4
+t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k
+t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp
+t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v
+w H))).
+
+theorem pr1_eta:
+ \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in
+(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl)
+(TLRef O) (lift (S O) O t))) t)))))
+\def
+ \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in
+(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift
+(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v
+(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H
+(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
+(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u))
+(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
+(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef
+O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O)
+u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind
+Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O)
+(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl
+(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u
+(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr
+not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O
+(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr2/props.ma".
+
+include "LambdaDelta-1/clen/getl.ma".
+
+theorem pr2_gen_ctail:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
+(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_:
+T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0
+(clen c) u t t2)))))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c)
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T
+(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda
+(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
+(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind
+Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3
+t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4)
+(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t))
+(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1)))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4
+t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda
+(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let
+H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind
+(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
+(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c)))
+(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
+(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
+T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d
+(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr)
+u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e:
+C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
+T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k
+u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c
+t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3
+t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4
+H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen
+c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
+(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq
+nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_:
+nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3
+T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
+(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8:
+(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T
+u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda
+(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0
+(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind
+Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
+t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
+t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda
+(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4
+(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1
+t2 H0))) H)))))).
+
+theorem pr2_gen_cbind:
+ \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
+(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1)
+(THead (Bind b) v t2)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c
+(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead
+(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y
+t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
+(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v
+t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead
+(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1
+(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
+c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u)
+v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d
+(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S
+j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead
+(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3)
+(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead
+d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d
+| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
+v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v)
+H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def
+(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def
+(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr
+(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t)))
+(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v
+(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7:
+(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c
+(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j)))
+(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b)
+v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S
+x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal
+nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i
+(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t
+(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead
+c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t
+H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
+
+theorem pr2_gen_cflat:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
+(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c
+(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2))
+(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2
+c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4
+H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
+c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u)
+v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0)))
+H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/defs.ma".
+
+include "LambdaDelta-1/getl/defs.ma".
+
+inductive pr2: C \to (T \to (T \to Prop)) \def
+| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to
+(pr2 c t1 t2))))
+| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2:
+T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1
+t)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr2/defs.ma".
+
+include "LambdaDelta-1/pr0/fwd.ma".
+
+include "LambdaDelta-1/getl/drop.ma".
+
+include "LambdaDelta-1/getl/clear.ma".
+
+theorem pr2_gen_sort:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to
+(eq T x (TSort n)))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
+T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda
+(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0
+t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0
+t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda
+(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t:
+T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n)))
+(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl
+i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda
+(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0
+t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0))
+(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n)
+(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n))))
+t1 H4))))))))))))) c y x H0))) H)))).
+
+theorem pr2_gen_lref:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to
+(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S
+n) O u)))))))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
+T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead
+d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O
+u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t)
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u))))))))))
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1
+t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda
+(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t:
+T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S
+n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n))
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u)))))))
+(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
+T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref
+t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef
+n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4)
+in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
+C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2
+(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in
+(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n))
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
+u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))
+(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O
+u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n))
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
+u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let
+H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr)
+u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))))
+(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S
+n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8)))
+(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
+
+theorem pr2_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y:
+T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in
+(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1
+x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1
+(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b:
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4))))))
+(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind
+Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2
+(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead
+(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1
+x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead
+(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead
+(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda
+(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind
+Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2
+x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2
+x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
+(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d
+u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0
+(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3:
+T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst)
+x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0
+x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead
+(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1
+x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2
+H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
+x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1
+t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t
+(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13:
+(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3)
+(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3
+(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2
+H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3
+H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9))))))))
+(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
+
+theorem pr2_gen_cast:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c
+t1 x))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
+t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0))))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0
+t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind
+T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast)
+x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T
+(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead
+(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2
+H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2
+H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3:
+(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5
+\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1)
+H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast)
+x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def
+(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1)
+H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead
+(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2:
+T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2
+c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
+x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12
+(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11))
+(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda
+(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat
+Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda
+(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12
+(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11))
+(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
+x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat
+Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0
+H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11))
+(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
+t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
+(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2
+H5)))))))))))))) c y x H0))) H))))).
+
+theorem pr2_gen_csort:
+ \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2)
+\to (pr0 t1 t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort
+n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_:
+C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind
+(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0
+t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
+t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c
+(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d
+(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr)
+u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
+
+theorem pr2_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
+t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
+u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3
+\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1)
+H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1
+x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
+(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl)
+x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat
+Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0
+H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2
+(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1
+x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind
+Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b:
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2
+H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
+(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b)
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat
+Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift
+(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0
+Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T
+t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda
+(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2
+x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
+x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0
+u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5
+H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
+t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T
+t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1
+x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
+T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T
+(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2:
+T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t
+(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T
+(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O
+u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i
+H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11:
+(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t
+(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
+(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat
+Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3:
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat
+Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr)
+u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2
+(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl)
+x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t
+H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
+x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13:
+(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1
+x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2
+(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl)
+x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1
+H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i
+H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind
+Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda
+(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2
+(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in
+(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
+(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2
+u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda
+(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead
+(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T
+(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0
+i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda
+(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2
+x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3
+H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t
+(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr)
+x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T
+(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
+(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d
+(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t
+(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15:
+(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5)
+(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1)
+t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
+(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0)
+(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12))
+(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda
+(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2)
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind
+x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl)
+(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1
+x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3:
+T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O
+x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
+(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3)
+x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t
+(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead
+(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead
+(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2:
+T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0)
+u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u
+x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
+(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3)
+x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6
+(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
+O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead
+(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
+(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6
+H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda
+(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3:
+T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda
+(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
+(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0)
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind
+x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0)
+x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat
+Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
+u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3)
+t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
+O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
+x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T
+(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2:
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
+(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T
+x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift
+(S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3)
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
+x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O
+x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22
+\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
+x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3:
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
+O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead
+(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
+(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4
+H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20)))))
+(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n
+i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead
+(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
+(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead
+(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
+(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
+(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19:
+(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat
+Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
+O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2)
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0
+u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S
+i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4)
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda
+(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u
+(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda
+(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
+T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0
+(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8)
+(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
+(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
+T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
+i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
+(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0)
+i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
+(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T
+(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
+O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead
+(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
+(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4
+H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead
+c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2
+x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S
+O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat
+Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14))
+(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl)
+(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t
+(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17:
+(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
+x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat
+Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
+u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3)
+t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
+O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
+x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T
+(\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2:
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
+(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T
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+(S O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3)
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
+x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O
+x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23
+\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
+x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
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+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
+O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
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+(Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead
+(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
+(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u
+i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8
+H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i
+(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T
+x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s
+(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7
+(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7)
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift
+(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
+x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3:
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2)
+t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
+Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0
+u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0
+(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6
+(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7
+H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
+(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0)
+i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
+(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0)
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda
+(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20:
+(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s
+(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
+(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
+(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
+T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
+i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
+(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind
+x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S
+O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in
+(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
+O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T
+(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl)
+(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta
+c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i)
+(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6)
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22)))))
+(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n
+i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5
+x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4
+(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6))
+(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
+
+theorem pr2_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
+T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
+(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
+t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
+Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
+(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2))))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
+t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in
+(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
+b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
+c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
+t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1
+t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr)
+u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2)))))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr)
+x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0
+x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda
+(y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
+T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (H7: (pr0 t1
+x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
+Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
+(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
+t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
+T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
+(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda
+(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
+(pr2_free c0 u1 x0 H6) (or3_intro0 (\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
+(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead
+c0 (Bind b) u) t1 x1 H7)))))) t2 H5)) (\lambda (H_x0: (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)))).(ex2_ind T (\lambda
+(y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
+Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
+(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
+t2))))) (\lambda (x2: T).(\lambda (H7: (pr0 t1 x2)).(\lambda (H8: (subst0 O
+x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
+Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
+(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
+t)))))) (ex2_ind T (\lambda (t: T).(subst0 O u1 x2 t)) (\lambda (t: T).(pr0 t
+x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
+T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))))
+(\lambda (x3: T).(\lambda (_: (subst0 O u1 x2 x3)).(\lambda (_: (pr0 x3
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
+T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
+(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda
+(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
+(pr2_free c0 u1 x0 H6) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
+(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
+(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1)) x0 H6 (pr2_delta (CHead c0 (Bind
+Abbr) x0) c0 x0 O (getl_refl Abbr c0 x0) t1 x2 H7 x1 H8))))))))
+(pr0_subst0_back x0 x2 x1 O H8 u1 H6)) t2 H5)))) H_x0)) H_x)))))) H4))
+(\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1
+t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr)
+u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))
+(\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 (lift (S
+O) O t2) H4))))) (pr0_gen_abbr u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d
+(Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
+t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
+(THead (Bind Abbr) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0
+t3 t2)) H2 (THead (Bind Abbr) u1 t1) H4) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t3))))))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0:
+T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3)))
+(ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1
+y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda
+(z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall
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+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x4: T).(\lambda (_: (subst0
+O u1 x2 x4)).(\lambda (_: (pr0 x4 x1)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0
+(Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0:
+T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3))))))) x3 x1 H13 (pr2_delta c0 d u i H1 u1 x0 H8 x3
+H14) (or3_intro1 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0)
+t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
+(Bind Abbr) u0) t1 x1))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x1))))
+(ex_intro2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
+(Bind Abbr) u0) t1 x1)) x0 H8 (pr2_delta (CHead c0 (Bind Abbr) x0) c0 x0 O
+(getl_refl Abbr c0 x0) t1 x2 H9 x1 H10)))))))) (pr0_subst0_back x0 x2 x1 O
+H10 u1 H8))))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t (THead
+(Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda
+(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0
+O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13
+(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_:
+T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z:
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr)
+u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl
+Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S
+i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl
+x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda
+(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
+(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t
+(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15:
+(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
+x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13
+(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
+u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O
+(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr)
+u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1
+x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8)))))))
+H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x))))))
+H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
+(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2)
+H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i)))))))
+(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
+
+theorem pr2_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift
+(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1
+t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind
+Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
+b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u)
+t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
+c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1
+x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1)
+(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
+c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1
+x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1
+H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0
+(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
+t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T
+t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1
+x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
+T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T
+(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t
+(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind
+Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8
+x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1
+x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead
+(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda
+(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0:
+T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0
+(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t
+(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14:
+(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11))
+(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
+t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S
+O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1
+t2 H5)))))))))))))) c y x H0))) H))))).
+
+theorem pr2_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
+(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1
+t2))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1)
+(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e)
+\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e
+t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e:
+C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2)))
+(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h
+d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T
+t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda
+(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1
+x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T
+t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda
+(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))
+x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5))))
+(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind
+Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
+t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
+(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def
+(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1
+x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h
+d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3)))
+(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def
+(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i
+(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d
+(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i))))
+(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d
+t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2:
+C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl
+i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0
+x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus
+i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind
+T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3)))
+(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h
+d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq
+T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0
+x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
+nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
+H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0
+t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h
+(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11
+H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1
+t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t
+(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13:
+(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d
+t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h)
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1
+H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1
+t2 h d H6)))))))))))))))) c y x H0))) H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr2/defs.ma".
+
+include "LambdaDelta-1/pr0/pr0.ma".
+
+include "LambdaDelta-1/getl/props.ma".
+
+theorem pr2_confluence__pr2_free_free:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0
+t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0
+t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t))
+(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2
+x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
+(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1)))))
+(pr0_confluence t0 t2 H0 t1 H))))))).
+
+theorem pr2_confluence__pr2_free_delta:
+ \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2)
+\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t))))))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0
+t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0
+t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4
+t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
+(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4:
+(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
+(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2
+c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2
+x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2:
+T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
+(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7:
+(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0
+H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u))))))
+(pr0_confluence t0 t4 H1 t1 H))))))))))))).
+
+theorem pr2_confluence__pr2_delta_delta:
+ \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall
+(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u:
+T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d
+(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c
+(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to
+(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t))))))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u:
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i
+c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0
+i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda
+(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t:
+T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4
+x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0
+t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2
+x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
+(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda
+(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0
+u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x
+x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))
+x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8))
+(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T
+(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind
+T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0:
+T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0
+t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
+(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H
+t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2))
+(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2
+w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2
+x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t:
+T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i
+i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0
+i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15:
+(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d
+u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9
+(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def
+(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15
+\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0)))
+H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0:
+C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind
+Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u)
+(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0
+(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono
+c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda
+(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x
+x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c
+(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0
+(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind
+(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t:
+T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda
+(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1
+x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in
+(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0
+(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T
+(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i
+u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i
+u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c
+t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2
+x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1
+x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u
+x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11)))
+(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10))
+(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x
+H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3
+H0))))))))))))))))))).
+
+theorem pr2_confluence:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
+(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
+t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in
+pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_:
+(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T
+(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with
+[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3:
+(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T
+t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c
+t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind
+T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5:
+T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4
+t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5:
+T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0
+t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t:
+T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0)
+\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6:
+T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda
+(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6
+t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5
+t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6
+t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7:
+T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t:
+T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7:
+T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0
+t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2
+H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) |
+(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1
+c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c
+(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d
+(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda
+(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14:
+(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c
+(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T
+(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))))
+(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d
+(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda
+(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16:
+(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda
+(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i
+H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1
+(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c)
+(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T
+t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4
+H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3
+t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0)
+\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4)
+\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda
+(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0
+(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to
+((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1
+t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind
+T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4)
+\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda
+(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12
+\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6:
+T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6
+t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2
+t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C
+c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c
+(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda
+(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7
+t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
+t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0
+t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
+t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1)
+(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6
+(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14
+H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow
+(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T
+t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to
+((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0
+t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
+t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T
+t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to
+((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda
+(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2
+(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to
+((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda
+(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr)
+u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6
+t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10
+H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1
+(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c)
+(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T
+t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C
+c) (refl_equal T t0) (refl_equal T t1)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr2/defs.ma".
+
+include "LambdaDelta-1/pr0/props.ma".
+
+include "LambdaDelta-1/getl/drop.ma".
+
+include "LambdaDelta-1/getl/clear.ma".
+
+theorem pr2_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0)))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0
+t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u
+(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0
+t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i
+H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0
+t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2
+u)))))))))))) c t1 t2 H)))))).
+
+theorem pr2_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0
+(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2
+t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1
+t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c
+u1 u2 H)))))).
+
+theorem pr2_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u)
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead
+k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c
+(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k
+u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3
+t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d:
+C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0))
+\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t:
+T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u
+t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4)
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u))
+\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1:
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4
+t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b)
+u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
+c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c
+(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind
+Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d
+(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr)
+u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in
+(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T
+u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda
+(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u)
+t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1:
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
+c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b)
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
+Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
+(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u
+t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n)
+(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3)
+(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead
+(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u)))))))))))))
+i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
+T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind
+Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall
+(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c
+(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0
+(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4:
+(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1:
+C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in
+(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c)
+(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f)
+u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4)
+(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t)
+(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1:
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
+c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f)
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
+Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
+(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u
+t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n)
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3)
+(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead
+(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u)))))))))))))
+i)))))) k) y t1 t2 H0))) H)))))).
+
+theorem clear_pr2_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1
+t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1:
+C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_:
+(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1:
+C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c
+(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
+
+theorem pr2_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free
+(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u
+i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c
+t1 t2 H)))))).
+
+theorem pr2_ctail:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0)
+t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail
+Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
+
+theorem pr2_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
+T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to
+(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
+(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind
+b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda
+(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2))
+(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2
+(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b)
+v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5:
+(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1:
+C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in
+(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind
+Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))
+(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr)
+u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
+v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind
+b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda
+(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind
+Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
+(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d
+(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr)
+u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in
+(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind
+T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def
+(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B
+Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match
+(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c
+(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda
+(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr)
+u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda
+(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda
+(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0)
+(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
+(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4)))))))))))))
+y t1 t2 H1))) H0)))))))).
+
+theorem pr2_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1
+t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c
+(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e)
+\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0
+e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d)))))))
+(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4
+t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1:
+C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def
+(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0
+(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9:
+(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11:
+(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i)
+(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i
+H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u
+H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h
+(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x
+(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S
+i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead
+x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d
+t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7
+h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h)
+(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3)
+(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h
+H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr2/defs.ma".
+
+include "LambdaDelta-1/pr0/subst1.ma".
+
+include "LambdaDelta-1/pr0/fwd.ma".
+
+include "LambdaDelta-1/csubst1/getl.ma".
+
+include "LambdaDelta-1/csubst1/fwd.ma".
+
+include "LambdaDelta-1/subst1/subst1.ma".
+
+include "LambdaDelta-1/getl/drop.ma".
+
+theorem pr2_delta1:
+ \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2)
+\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2
+t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0)
+(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2
+H0 t0 H2))) t H1)))))))))).
+
+theorem pr2_subst1:
+ \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1
+t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T
+(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))
+(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1:
+T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda
+(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1:
+T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T
+(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))))
+(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))
+(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))
+(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4
+x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v
+t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v
+(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda
+(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda
+(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c
+(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2:
+T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda
+(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda
+(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1
+x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2:
+T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not
+(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0:
+T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
+T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t
+x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x
+H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t
+H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let
+H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in
+(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind
+Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v)
+(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead
+e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal
+C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v)
+(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d
+(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _
+t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u)
+(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in
+(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl
+i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u
+(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C
+d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in
+(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x
+t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t
+w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21:
+(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
+T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20))))
+(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9)))))))
+H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0
+H5))))))))))))))) y t1 t2 H1))) H0)))))))).
+
+theorem pr2_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
+(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
+x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e:
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to
+(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0
+a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda
+(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1
+x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
+d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d
+x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2:
+T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
+x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0
+(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda
+(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T
+(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
+x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda
+(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t))
+H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4
+(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0
+H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S
+O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e:
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1:
+T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda
+(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2))
+(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2:
+T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1)
+x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x
+(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2:
+T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2)))
+(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10:
+(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0))
+H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1
+d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12:
+(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0:
+T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0
+t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2:
+T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O)
+d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d
+(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3:
+T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3)))
+(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr)
+u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i)
+(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0
+(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr)
+d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda
+(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_:
+C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0
+t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4:
+T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr)
+x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1
+(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl
+i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0
+(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i))))
+(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0
+u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6:
+T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i))
+x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop
+(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0:
+T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6)
+H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i))
+x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S
+i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0
+x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28:
+(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda
+(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind
+nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S
+O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in
+(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S
+i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9:
+T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9)))
+(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S
+i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8
+(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift
+(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat
+(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S
+i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let
+H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n:
+nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12))
+in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10)))
+(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O)
+d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32))))))))
+(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30))))))
+(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S
+i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i
+H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12
+c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i
+(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12))))
+(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n:
+nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def
+(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15
+\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in
+(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind
+Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2:
+T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let
+H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1))
+H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead
+e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _
+_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)
+(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in
+((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
+(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind
+Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d
+e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind
+Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0:
+T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r
+T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u
+(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2)))
+(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1:
+C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda
+(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0)
+t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2:
+T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t
+x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind
+T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0)
+(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i)
+(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i
+(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i
+x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10))))))
+(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0)
+H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T
+(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S
+O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2)))
+(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0
+u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T
+(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1
+(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O)
+d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq
+T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0
+x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13
+(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t
+(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u
+(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0
+(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d
+(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n:
+nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3
+H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S
+O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S
+O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift
+(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12))))))))))
+(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0
+x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr2/defs.ma".
+
+inductive pr3 (c: C): T \to (T \to Prop) \def
+| pr3_refl: \forall (t: T).(pr3 c t t)
+| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3:
+T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/props.ma".
+
+include "LambdaDelta-1/pr2/fwd.ma".
+
+theorem pr3_gen_sort:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to
+(eq T x (TSort n)))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t:
+T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda
+(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t:
+T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2:
+T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
+(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3
+t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda
+(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t:
+T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n))
+\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T
+(TSort n)))) t1 H4))))))))) y x H0))) H)))).
+
+theorem pr3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
+T).(pr3 (CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y:
+T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead
+(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0:
+T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c
+(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t
+(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda
+(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0
+x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl
+(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda
+(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda
+(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1))
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
+(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
+t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1
+t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
+(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0
+x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4:
+T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5
+t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3
+(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
+(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst)
+x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))
+x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u:
+T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u))))))))))
+H11)))))))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_cast:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c
+t1 x))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1
+t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T
+t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall
+(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0
+t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall
+(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0
+x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
+x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c
+x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T
+(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1)))))
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1:
+T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4:
+(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t:
+T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def
+(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c
+x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2
+c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda
+(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5))
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast)
+x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2
+x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T
+t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16:
+(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat
+Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead
+(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5
+(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15)
+(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3
+t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c
+x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4
+H2))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
+(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1
+t2))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1)
+(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e)
+\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e
+t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda
+(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T
+(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2)))))))
+(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h
+d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T
+t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t:
+T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e:
+C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h
+d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda
+(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
+T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2
+(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5:
+T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda
+(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5:
+(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1
+(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in
+(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e
+x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5:
+T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d
+x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift
+h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4
+(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda
+(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T
+(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2
+H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x
+H0)))) H)))))).
+
+theorem pr3_gen_lref:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to
+(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T x (lift (S n) O v))))))))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t:
+T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y
+x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or
+(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T
+t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t))))
+(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3:
+T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or
+(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef
+n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4)
+in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
+v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef
+n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T
+t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef
+n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or
+(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def
+(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8
+(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d:
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n))
+(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead
+d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
+v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0
+(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def
+(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C
+T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
+v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda
+(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift
+c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda
+(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq
+T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T
+t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3
+(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v:
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3
+(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13)))))
+H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))).
+
+theorem pr3_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1
+(lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c
+(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y
+x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1
+(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c
+(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t
+(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1
+(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind
+Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1
+(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c
+(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1))
+(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b)
+u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c
+t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall
+(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let
+H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0
+x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O
+t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind
+Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
+(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq
+T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O)
+O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind
+Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind
+Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2)))
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O
+t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3
+(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c
+(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c
+(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4
+x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c
+(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14
+(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead
+c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13:
+(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind
+Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift
+(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7:
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O
+t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))
+(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S
+O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void)
+O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr)
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda
+(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O)
+O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind
+Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3
+(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t:
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind
+Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda
+(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0
+x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0)
+x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0
+x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1)))))
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1:
+T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
+(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
+t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1
+t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u:
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z:
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0)
+z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1
+(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead
+c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z
+t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u:
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z:
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0)
+z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2
+c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c
+(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2
+(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_:
+T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z:
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0)
+z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c
+(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3))))
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind
+Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3
+(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2
+x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3)))
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind
+Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c
+(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c
+x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T
+(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4
+x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1
+(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4
+H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr)
+x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O)
+O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u:
+T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1
+x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c
+(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4:
+T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4)
+x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall
+(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda
+(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5
+x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr)
+x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0)
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5
+x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0)
+(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3
+x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c
+(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1
+x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O
+t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
+(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq
+T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1
+x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0)
+x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall
+(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda
+(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6
+x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr)
+x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0)
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6
+x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1
+H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind
+Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7
+(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17))
+(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr)
+x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4
+(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing
+(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3
+(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10))))))
+H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing
+(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4)
+(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c
+(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1
+t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1
+(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda
+(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall
+(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))))
+(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t
+(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda
+(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat
+Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0
+x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda
+(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2
+t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat
+Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
+z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
+(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
+t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1
+t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2
+x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t
+(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
+(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t:
+T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3
+(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4
+x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T
+(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4
+x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl)
+x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c
+x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T
+(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10
+x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15:
+(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda
+(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro
+T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5:
+T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1
+t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10
+(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3
+c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9
+(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2
+x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9)
+x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))
+x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17
+(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7))
+(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq
+T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4
+x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2
+x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t
+(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
+(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t:
+T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind
+Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u)
+x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6:
+T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T
+x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7
+(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0
+x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7)
+x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2
+(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl)
+x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
+(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let
+H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7
+(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4))
+H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2)
+x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1:
+T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind
+b) u1) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
+(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind
+b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3
+(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B
+Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x:
+T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def
+(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr)
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1)
+t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1)
+t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x
+(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3
+(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1
+H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
+O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H:
+(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1
+\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c
+(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_:
+(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1
+\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c
+(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void)
+u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda
+(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void)
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
+u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead
+c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0
+x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall
+(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1
+H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1
+(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/fwd.ma".
+
+include "LambdaDelta-1/iso/props.ma".
+
+include "LambdaDelta-1/tlist/props.ma".
+
+theorem pr3_iso_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat
+Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w))
+u2))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
+(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
+(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2:
+T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to
+(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq
+T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda
+(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso
+(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T
+(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef
+i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T
+(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind
+Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O
+v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0
+(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift
+(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t)
+\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S
+i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C
+(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind
+Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1)
+H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
+c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d
+(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead
+x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind
+Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1
+(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13
+\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def
+(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d
+H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d
+H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15)))))))
+H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef
+i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P:
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w))
+u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3
+\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3
+t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda
+(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t
+x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def
+(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
+t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1)
+H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t
+x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat
+Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1)))
+u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t
+u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
+w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t
+x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c
+(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t
+(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
+(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead
+(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P:
+Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead
+(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1))
+(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst)
+x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1
+(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t
+x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5))))))))))
+H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat
+Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda
+(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat
+Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0)
+x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind
+x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c
+(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1
+x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0
+(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P:
+Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat
+Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift
+(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2))
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0
+H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind
+x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1
+(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead
+(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift
+(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O
+(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2
+(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2))))
+u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12
+c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat
+Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10
+(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
+
+theorem pr3_iso_appls_cast:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1
+\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2:
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c
+(THeads (Flat Appl) vs t) u2))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0
+(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2))))))
+(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda
+(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1
+\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
+t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0
+x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def
+(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall
+(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat
+Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat
+Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2:
+(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H:
+((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2)
+\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall
+(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2:
+T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead
+(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P:
+Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead
+(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda
+(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat
+Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2)
+(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2
+(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2)
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl)
+x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl)
+t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2:
+T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v
+t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in
+(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat
+Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl)
+t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3))
+(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1
+t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))
+(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c
+(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6:
+(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0
+x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b)
+u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1))
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads
+(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6
+(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead
+(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast
+Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c
+(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1)
+(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind
+Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c
+t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3:
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b)
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat
+Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t))
+(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda
+(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat
+Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t))
+c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0
+(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead
+(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads
+(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda
+(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0
+Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2
+c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead
+(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead
+(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0)
+x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2
+(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4)
+x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift
+(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead
+c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind
+x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7)
+(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift
+(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
+O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
+(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2)
+(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5)
+x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
+
+theorem pr3_iso_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2:
+T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t))
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O v1) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
+(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c
+(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead
+(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let
+H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T
+T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1
+t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst))))))))
+(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1
+z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq
+T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_:
+(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0:
+T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P:
+Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl)
+x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S
+O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl))
+(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead
+(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1
+t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u:
+T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr)
+x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b)
+v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall
+(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3)
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def
+(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in
+(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))))
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind
+Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
+T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda
+(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4
+x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2)
+t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return
+(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow
+Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst)
+x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0
+| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind
+Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0]))
+(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T
+x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda
+(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def
+(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def
+(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst
+H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H
+Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22
+\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_:
+False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t))
+(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13)))))))
+H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind
+Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
+x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift
+(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead
+(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O
+v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop
+(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst)
+x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead
+(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead
+(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2
+x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst)
+x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2
+(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead
+(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta
+x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3)
+(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead
+(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1)))
+(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4)))))))))
+H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3:
+T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O)
+O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0)
+y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
+v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0
+Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1
+x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
+O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1
+x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind
+x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t
+(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind
+b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat
+Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl)
+(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq
+T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2
+x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal
+T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match
+k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
+((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0
+_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
+((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
+(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def
+(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17)
+in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18)
+in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2
+x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0
+Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat
+Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8)
+(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S
+O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O
+x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c
+(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3
+(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3
+(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11:
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
+(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1)
+t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat
+Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b)
+(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift
+(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2)
+v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind
+x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O
+(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c
+(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
+x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O
+x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind
+x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl)
+x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
+O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2))
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0
+H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5
+H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl)
+(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S
+O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
+(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O))))
+H10))) u2 H6))))))))))))) H3)) H2)))))))))).
+
+theorem pr3_iso_appls_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u:
+T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs
+(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2:
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c
+(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
+t))) u2)))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda
+(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
+b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead
+(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c:
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b)
+u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t))
+u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1)))))
+(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall
+(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
+t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
+b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda
+(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda
+(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v
+(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def
+(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
+b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
+T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead
+(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
+Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))))
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat
+Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat
+Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl)
+t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P:
+Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl)
+x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8
+(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
+t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat
+Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T
+T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3)
+u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1
+(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0
+x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind
+b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0
+(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
+t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S
+O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u
+(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c
+(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead
+(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P:
+Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t)
+t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl)
+t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead
+(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta
+x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr)
+x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3
+(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
+t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
+Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda
+(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c
+(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
+O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u
+(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead
+(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl)
+t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t
+(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c
+(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2)
+H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
+(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P:
+Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1
+H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0)
+x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind
+x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat
+Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl)
+(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl
+x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
+O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat
+Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2)
+(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4)
+(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c
+(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind
+x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5
+(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat
+Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat
+Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3)
+(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2
+H7)))))))))))))) H4)) H3))))))))) vs)))))).
+
+theorem pr3_iso_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u:
+T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t))
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl)
+(lifts (S O) O vs) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs:
+TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0:
+T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c:
+C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P:
+Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t)
+t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda
+(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso
+(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts:
+TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0:
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead
+(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0))
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat
+Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0:
+T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl)
+(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat
+Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P:
+Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t))
+(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0))
+u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl)
+(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let
+H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0))
+(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl)
+t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0)
+ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind
+b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2
+(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0)))
+(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda
+(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall
+(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to
+((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P:
+Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O
+t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t
+(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat
+Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c
+(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl)
+(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1:
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead
+(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0
+t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads
+(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c
+(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0)))
+u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b
+H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_:
+((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3
+c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads
+(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to
+(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2))
+u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
+(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead
+(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift
+(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2:
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1
+ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1
+ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0
+(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2))
+u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead
+(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to
+(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2
+(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8:
+(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads
+(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O
+t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
+(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads
+(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat
+Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t))
+(lifts_tapp (S O) O t ts))))))))))) vs))).
+
+theorem pr3_iso_beta:
+ \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat
+Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c
+u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind
+Abbr) v t) u2))))))))
+\def
+ \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2:
+T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t))
+u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2)
+\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind
+Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
+T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2))))
+(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
+T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
+w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
+b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c
+(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst)
+w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v
+x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T
+u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0)
+\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T
+(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t)
+t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead
+(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2:
+(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
+T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
+w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
+b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v
+u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t)
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v
+x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0
+x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b)
+u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5)
+in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0]))
+(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0)
+\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in
+(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1
+H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0
+H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c
+(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2)
+(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2))
+(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind
+Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5
+(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v
+x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2
+x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in
+(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
+(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead
+(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead
+(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
+\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2)
+(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 |
+(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0)
+x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1
+x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda
+(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0))))
+H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0))
+H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind
+b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b:
+B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2))
+H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b
+Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in
+False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with [])
+in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
+
+theorem pr3_iso_appls_beta:
+ \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1
+\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in
+(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr)
+v t)) u2)))))))))
+\def
+ \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall
+(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl)
+v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1
+u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat
+Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w:
+T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P:
+Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1:
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead
+(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl)
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P:
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))
+u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c:
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1:
+(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v
+(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def
+(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
+Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl)
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2)
+(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))
+(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)))
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat
+Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P:
+Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl)
+x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0
+(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat
+Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T
+T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat
+Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c
+(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3
+c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))
+(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u:
+T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1)
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
+(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1
+c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0
+x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1
+(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1)
+(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr)
+t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead
+(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t
+x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3:
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst)
+w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead
+(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq
+B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v
+(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda
+(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c
+(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
+O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr)
+v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2))
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
+(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c
+(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead
+(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda
+(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst)
+w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O)
+O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead
+(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl)
+(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1)
+x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O
+x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl)
+(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12
+(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c
+(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7)
+(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift
+(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
+O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
+(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2)
+(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5)
+x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/defs.ma".
+
+include "LambdaDelta-1/pr1/defs.ma".
+
+theorem pr3_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t:
+T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0
+t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c:
+C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/props.ma".
+
+include "LambdaDelta-1/pr2/pr2.ma".
+
+theorem pr3_strip:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
+(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2
+t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda
+(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2
+t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t:
+T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda
+(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t:
+T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c
+t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t))
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c
+x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0))))))))))
+t0 t1 H)))).
+
+theorem pr3_confluence:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
+(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2
+t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda
+(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda
+(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4
+t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c
+t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2
+t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)))
+(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t))
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5
+c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1
+H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/pr1.ma".
+
+include "LambdaDelta-1/pr2/props.ma".
+
+include "LambdaDelta-1/pr1/props.ma".
+
+theorem clear_pr3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t:
+T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3
+t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
+
+theorem pr3_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
+
+theorem pr3_t:
+ \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall
+(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0
+t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3
+c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall
+(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3:
+(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+theorem pr3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0:
+T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t:
+T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0
+t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u
+t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c
+t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
+
+theorem pr3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall
+(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda
+(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2:
+T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
+(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c
+(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing
+c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t)
+(H2 k t)))))))))) u1 u2 H)))).
+
+theorem pr3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u)
+(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0))))
+(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u
+t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0)
+(THead k u t4) H2))))))) t1 t2 H)))))).
+
+theorem pr3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
+(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c
+u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
+
+theorem pr3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
+(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c
+u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pr3_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v:
+T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f:
+F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead
+c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c
+(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
+
+theorem pr3_flat:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead
+(Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
+(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f
+u2))))))))).
+
+theorem pr3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
+t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_:
+(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k
+u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4:
+(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead
+c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d
+(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t))))
+(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8:
+(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d
+(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda
+(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind
+Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2
+(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b)
+u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
+u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind
+b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr
+b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0:
+T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3
+(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4
+t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t)
+(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x
+t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr)
+u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind
+Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17))))))
+(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f:
+F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr)
+u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O
+(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c
+(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind
+Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda
+(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4
+t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k
+u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind
+(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to
+((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t)
+\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda
+(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr)
+u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2
+(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0)
+(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
+(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f:
+F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr)
+u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2
+(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0)
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f
+u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
+
+theorem pr3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
+u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3
+(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
+K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
+(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1:
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y:
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5
+(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
+c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n:
+nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
+t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3
+(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
+(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0
+| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind
+b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t)
+(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
+Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr)
+u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let
+H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in
+(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind
+T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i
+O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda
+(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let
+H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O
+i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n
+u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7))
+(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda
+(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0
+x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind
+Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead
+c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i)
+(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1)
+(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x
+O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14))
+H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0
+(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6
+(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0
+(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5
+H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0)
+H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0
+(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4
+t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr)
+u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl
+(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1)
+t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t)
+(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta
+(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind
+Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4
+t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat
+f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6
+(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f)
+c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10)))))
+i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))).
+
+theorem pr3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to
+(pr3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
+u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2)
+\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1
+u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
+u1 H3)))))))))) t1 t2 H)))))).
+
+theorem pr3_pr3_pr3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
+t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3
+(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0:
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0
+t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
+
+theorem pr3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1
+t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h
+d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda
+(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0
+t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d
+t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2
+H0)))))))).
+
+theorem pr3_eta:
+ \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind
+Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v
+(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
+\def
+ \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind
+Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead
+(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c
+(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w
+u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead
+(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat
+Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u
+(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat
+Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)))
+(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S
+O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing
+(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O)
+(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O))
+(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))
+(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind
+Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u
+(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead
+(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/defs.ma".
+
+include "LambdaDelta-1/pr2/subst1.ma".
+
+theorem pr3_subst1:
+ \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2)
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0:
+T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1
+w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1:
+T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1
+w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v
+t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i
+v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T
+(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T
+(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2)))
+(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3
+x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5
+w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5
+w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v
+t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1
+i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c
+e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
+
+theorem pr3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
+(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a
+x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
+(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1
+d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)))))))))))))))
+(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
+(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_:
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0
+a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d
+x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2)))
+(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4:
+T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
+(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2:
+T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1
+x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
+(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4:
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0
+a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d
+x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda
+(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
+x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1
+d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda
+(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2))
+(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2:
+T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O)
+d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d
+u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x
+x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0
+e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/props.ma".
+
+include "LambdaDelta-1/wcpr0/getl.ma".
+
+theorem pr3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1:
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0
+t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2:
+T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1)
+(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t:
+T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
+(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead
+c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3
+(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0))
+(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7:
+(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq
+C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c:
+C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_:
+(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0
+k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda
+(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k
+u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind
+Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2:
+C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3:
+T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr)
+x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T
+(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3
+(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6
+x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta
+(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t
+(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1
+H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3
+H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7)))
+H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2
+H2)))))))))))))) c2 c1 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Base-1/theory.ma".
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+definition r:
+ K \to (nat \to nat)
+\def
+ \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i |
+(Flat _) \Rightarrow (S i)])).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/r/defs.ma".
+
+include "LambdaDelta-1/s/defs.ma".
+
+theorem r_S:
+ \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
+i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
+f) i))))) k).
+
+theorem r_plus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
+(plus (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
+(Flat f) i) j))))) k).
+
+theorem r_plus_sym:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
+(plus i (r k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
+F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
+
+theorem r_minus:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
+(minus (r k i) (S n)) (r k (minus i (S n)))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
+K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
+n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
+F).(minus_x_Sy i n H)) k)))).
+
+theorem r_dis:
+ \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
+\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i:
+nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i)
+(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H:
+((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_:
+((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i:
+nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
+((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
+((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
+(i: nat).(refl_equal nat (S i)))))))) k).
+
+theorem s_r:
+ \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0
+i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i))))
+(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
+
+theorem r_arith0:
+ \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
+\def
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
+nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
+nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
+(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
+
+theorem r_arith1:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
+i)) (S j)) (minus (r k i) j))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
+(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
+(minus (r k i) j)) (r k (S i)) (r_S k i)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+definition s:
+ K \to (nat \to nat)
+\def
+ \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) |
+(Flat _) \Rightarrow i])).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/s/defs.ma".
+
+theorem s_S:
+ \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S
+i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat
+f) i))))) k).
+
+theorem s_plus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
+(plus (s k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j)))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s
+(Flat f) i) j))))) k).
+
+theorem s_plus_sym:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
+(plus i (s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq
+nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j))
+(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j:
+nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
+
+theorem s_minus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s
+k (minus i j)) (minus (s k i) j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j))))))
+(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j
+i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i)
+j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H))))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j
+i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
+
+theorem minus_s_s:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s
+k j)) (minus i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_:
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j)))))
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i
+j))))) k).
+
+theorem s_le:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i)
+(s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
+nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_:
+F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
+
+theorem s_lt:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i)
+(s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
+nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda
+(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
+
+theorem s_inj:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j))
+\to (eq nat i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s
+(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda
+(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
+
+theorem s_inc:
+ \forall (k: K).(\forall (i: nat).(le i (s k i)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i))))
+(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s
+(Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n
+(s (Flat f) i)))) k).
+
+theorem s_arith0:
+ \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
+\def
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n:
+nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal
+nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
+
+theorem s_arith1:
+ \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
+\def
+ \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n
+i)) (refl_equal nat i) (minus i O) (minus_n_O i))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubc/arity.ma".
+
+include "LambdaDelta-1/csubc/getl.ma".
+
+include "LambdaDelta-1/csubc/drop1.ma".
+
+include "LambdaDelta-1/csubc/props.ma".
+
+theorem sc3_arity_csubc:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall
+(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
+C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_:
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T
+(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0)))
+(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2
+n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n
+is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall
+(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
+a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda
+(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let
+H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2:
+C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u))))
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1
+(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr)
+(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x
+(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def
+H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2:
+C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2
+(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2
+x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u))
+x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind
+Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0
+(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
+(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2)))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))))
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq
+C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc
+g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1
+(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is
+(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr)
+(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind
+C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr)
+(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in
+(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y
+(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O
+u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i)
+O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O)
+(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4)
+(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans
+is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef
+i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
+w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq
+K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is
+(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13:
+(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr)
+x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1
+(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0
+(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14)
+in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13)
+in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12))
+(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b:
+B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2)))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq
+K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_:
+(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is
+i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind
+Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 (lift1
+is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) H5))))))))))))))))
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1:
+(arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
+(asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is:
+PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g
+d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 H3 Abst d
+u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is
+i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1
+(ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x:
+C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is i)
+d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def
+(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is
+i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans
+is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans
+is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda
+(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst)
+(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1
+(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C
+(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u))))
+(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
+w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
+x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
+T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2
+C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u))))
+(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0
+(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
+c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C
+x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x
+x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0))
+H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def
+(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0:
+T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef
+(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1
+t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0
+H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1
+(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is
+i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
+w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq
+K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is
+(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_:
+(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr)
+x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x
+(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def
+(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind
+Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef
+(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2
+(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let
+H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in
+(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S
+(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g
+x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0)
+H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13))
+(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
+T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
+C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b:
+B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2)))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq
+K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_:
+(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is
+i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind
+Abst) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 (lift1
+is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) H6)))))))))))))))))
+(\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2:
+((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
+C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0
+a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1
+(CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2
+(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H5:
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g d1 c2)).(let H_y
+\def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead (Bind b) (lift1 is u)
+(lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u)
+(lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) (Ss is)
+(drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) (csubc_head
+g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is (THead
+(Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g
+a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is
+u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g
+d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is:
+PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g
+d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))
+(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall
+(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g
+a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead
+(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d:
+C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d
+c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1
+is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind
+Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1
+is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2
+(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst)
+(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d:
+C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0:
+PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1
+is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3
+g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8
+d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr
+(\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) a1 a2 TNil) in (H_y d w
+(lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def (csubc_drop1_conf_rev g is0
+d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C (\lambda (c3: C).(drop1 is0
+c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w)
+(lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: C).(\lambda (H10: (drop1
+is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T (lift1 (papp (Ss is0)
+(Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind Abbr) w) t1))
+(eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g a2 (CHead d
+(Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 (papp is0 is)
+u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x u (drop1_trans
+is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x d H11 (lift1
+(papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 H10 is c H4) x
+(csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 is)) (lift1 (Ss
+is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) t0))))) H9))) H6)) H6
+(lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) is0 d (lift1 is u) (H1
+d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss
+is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 (Ss is) t0))))))))) (lift1
+is (THead (Bind Abst) u t0)) (lift1_bind Abst is u t0))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u
+a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is
+u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0
+(AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: PList).((drop1
+is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2
+(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4:
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y
+\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6
+\def H_y0 in (land_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d:
+C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d
+c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0)))))))))
+(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2
+(lift1 is t0) (AHead a1 a2))).(\lambda (H8: ((\forall (d: C).(\forall (w:
+T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2
+d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))))).(let H_y1 \def (H8
+c2 (lift1 is u) H_y PNil) in (eq_ind_r T (THead (Flat Appl) (lift1 is u)
+(lift1 is t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2))
+(lift1 is (THead (Flat Appl) u t0)) (lift1_flat Appl is u t0)))))
+H6)))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
+A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (d1:
+C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1
+c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda
+(_: (arity g c t0 a0)).(\lambda (H3: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a0
+c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4:
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y
+\def (sc3_cast g a0 TNil) in (eq_ind_r T (THead (Flat Cast) (lift1 is u)
+(lift1 is t0)) (\lambda (t1: T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1
+is H4 c2 H5) (lift1 is t0) (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast)
+u t0)) (lift1_flat Cast is u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0:
+T).(\lambda (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall
+(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g
+d1 c2) \to (sc3 g a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2:
+(leq g a1 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is
+d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2
+(lift1 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
+
+theorem sc3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (sc3 g a c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y
+(drop1_nil c) c (csubc_refl g c))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sn3/defs.ma".
+
+include "LambdaDelta-1/arity/defs.ma".
+
+include "LambdaDelta-1/drop1/defs.ma".
+
+definition sc3:
+ G \to (A \to (C \to (T \to Prop)))
+\def
+ let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c:
+C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t
+(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead
+a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is
+t)))))))))]))) in sc3.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sc3/defs.ma".
+
+include "LambdaDelta-1/sn3/lift1.ma".
+
+include "LambdaDelta-1/nf2/lift1.ma".
+
+include "LambdaDelta-1/csuba/arity.ma".
+
+include "LambdaDelta-1/arity/lift1.ma".
+
+include "LambdaDelta-1/arity/aprem.ma".
+
+include "LambdaDelta-1/llt/props.ma".
+
+include "LambdaDelta-1/drop1/getl.ma".
+
+include "LambdaDelta-1/drop1/props.ma".
+
+include "LambdaDelta-1/lift1/props.ma".
+
+theorem sc3_arity_gen:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c
+t) \to (arity g c t a)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind
+(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
+t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity
+g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_:
+(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to
+(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity
+g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in
+(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
+Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity
+g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
+Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
+
+theorem sc3_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c
+t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c:
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3
+g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3:
+A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to
+(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c:
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3
+g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall
+(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3
+c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda
+(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c
+t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0
+in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda
+(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
+(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n
+n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k)
+(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
+(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0:
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort
+n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1
+x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in
+(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0)
+H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity
+g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2))))))))))
+(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c:
+C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
+(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to
+(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0:
+A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c:
+C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
+(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t)
+\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1:
+((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t:
+T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c
+t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t
+(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is
+t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4
+\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
+T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
+(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity
+g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a
+d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat
+Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in
+(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a
+a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4:
+A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0
+x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A
+(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1)
+(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall
+(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d
+c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t
+(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d:
+C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is:
+PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14:
+(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0
+t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0
+(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat
+Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0)
+(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3
+H11))))))) H7))))) H4)))))))))))) a2)) a1)).
+
+theorem sc3_lift:
+ \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e
+t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
+\to (sc3 g a c (lift h d t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e:
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t))))))))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda
+(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in
+(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
+(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n
+n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0))
+(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e
+t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e:
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d
+t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t:
+T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e:
+C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall
+(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
+e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3
+\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall
+(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g
+a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
+(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
+(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
+(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda
+(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
+PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
+t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0:
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t)))))))))
+(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w:
+T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1
+is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1
+(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w
+t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t))
+(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
+
+theorem sc3_lift1:
+ \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds:
+PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e)
+\to (sc3 g a c (lift1 hds t)))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g
+a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c:
+C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c
+e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0:
+C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0:
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3
+g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c:
+C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n
+n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x
+in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2
+e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n
+n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0
+H4) c n n0 H3)))) H2))))))))))) hds)))).
+
+theorem sc3_abbr:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i:
+nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads
+(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to
+(sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
+TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
+C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
+(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef
+i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs:
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c:
+C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v))
+(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g
+c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat
+Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef
+i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2:
+(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda
+(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c
+(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs
+(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2)
+(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda
+(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v:
+T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to
+((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs
+(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs:
+TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
+C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
+(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef
+i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda
+(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs
+(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0
+d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat
+Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda
+(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity
+g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0:
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
+(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead
+a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
+PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
+(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads
+(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0:
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
+(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i))
+(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
+(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
+(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs
+(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0
+w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def
+(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C
+(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is
+i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x:
+C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i)
+d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w
+(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is
+(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r
+T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w
+(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans
+is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1
+d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T
+(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1
+d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1
+is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v)
+vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v))
+H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs
+(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8)))))))))))
+H3))))))))))))) a)).
+
+theorem sc3_cast:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
+(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t:
+T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl)
+vs (THead (Flat Cast) u t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
+TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat
+Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to
+(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
+T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) |
+(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t:
+T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0))
+(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g
+(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
+(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads
+(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land
+(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0))
+(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1:
+(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2:
+(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat
+Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs
+u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads
+(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads
+(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat
+Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs
+t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads
+(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat
+Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat
+Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs
+t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort
+O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))
+(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t
+H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with
+[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c
+(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t)
+(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads
+(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1
+n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads
+(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let
+H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))
+(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs
+(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs
+(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u)
+(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def
+H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3
+c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead
+(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead
+(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort
+(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g
+c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs
+(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n
+H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall
+(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to
+(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1:
+A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3
+g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c
+(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead
+(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
+T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc
+g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
+is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land
+(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
+d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3
+\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
+a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
+is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs
+(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3
+g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u
+t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0
+(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d
+w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2
+in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
+t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity
+g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
+t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c
+u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9:
+(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y
+\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1
+is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d
+(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1
+is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl)
+(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat
+Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w
+t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u))
+(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat
+Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w
+H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl
+is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t))
+(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl
+is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
+
+theorem sc3_props__sc3_sn3_abst:
+ \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g
+a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def
+(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
+((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c:
+C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs:
+TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in
+(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
+(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall
+(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3
+c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
+(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to
+((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n
+n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c:
+C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
+t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
+t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2))
+H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H:
+(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0:
+(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat
+Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H
+(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land
+(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall
+(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl)
+vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c
+(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0:
+(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t))))
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
+(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
+(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c:
+C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t))))
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
+(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c
+vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))
+(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads
+(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t:
+T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
+d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind
+(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0))))
+(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads
+(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to
+(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_:
+((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0
+t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
+C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i))
+\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef
+i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0:
+T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i:
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to
+((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs
+(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0:
+T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs:
+TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs
+(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0
+(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind
+(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
+\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
+(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0
+a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
+(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
+(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0)
+in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2
+O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10
+(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1)
+(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0
+H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1))
+I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1)
+(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0
+(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil
+(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst)
+x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3
+(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S
+x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef
+O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O
+t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop
+(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2)))))
+(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g
+c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c
+(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl)
+vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
+\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
+(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d:
+C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is:
+PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall
+(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
+TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
+vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0
+c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl)
+w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0:
+C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_:
+((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0
+(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3
+c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9
+\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to
+(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0:
+C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef
+i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef
+i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
+(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t)
+\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0:
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1)
+\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat
+Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs)))
+in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i)))
+(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef
+(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat
+Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i))
+(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
+is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i)))
+(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0
+(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1
+(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1))
+(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is
+(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is
+(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2)
+(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is
+vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i))
+(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat
+Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
+
+theorem sc3_sn3:
+ \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c
+t) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
+(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
+H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
+c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g
+c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0
+vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t)
+(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0
+t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
+C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i))
+\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef
+i))))))))))).(H1 c t H))) H0))))))).
+
+theorem sc3_abst:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
+(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef
+i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda
+(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i))
+a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
+(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0:
+C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
+TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
+vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a
+c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl)
+vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t)
+\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0:
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to
+((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl)
+vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))).
+
+theorem sc3_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1:
+A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v:
+T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts
+(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs
+(THead (Bind b) v t)))))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
+(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall
+(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads
+(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads
+(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0:
+nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t:
+T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat
+Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0
+in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O
+vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S
+O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t))
+(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda
+(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
+(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind
+b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))
+(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0)
+H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2))))))))))
+(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall
+(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl)
+vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall
+(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead
+c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v)
+\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v
+t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda
+(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a
+d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g
+a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs)
+t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind
+(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
+(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
+(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat
+Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land
+(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0))
+(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
+(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c
+(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda
+(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
+PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl)
+w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity
+g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d:
+C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead
+(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1
+H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3
+g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def
+(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is
+vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead
+(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is)
+t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl)
+(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList
+(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d
+(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat
+Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl)
+(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is
+v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is
+v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S
+O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is)
+(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts
+(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O
+vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is
+d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is
+(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead
+(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
+
+theorem sc3_appl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs:
+TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads
+(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w:
+T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead
+(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a:
+A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3
+g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v)
+\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat
+Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v:
+T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs
+(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead
+(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda
+(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads
+(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat
+Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs
+(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads
+(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3:
+(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n
+n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v
+t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead
+(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat
+Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen
+g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3)
+(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1)))))
+H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall
+(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs
+(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g
+(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v
+(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall
+(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c
+(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to
+(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl)
+vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs:
+TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land
+(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0))
+(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
+(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c
+v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1
+in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))
+(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
+(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c
+(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead
+a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
+(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w
+t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind
+Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0:
+T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
+(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v
+t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v
+(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0:
+T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
+(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v
+(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g
+c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5)
+(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is:
+PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1
+is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda
+(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat
+Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3
+g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0))))
+(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0:
+T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs)
+(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1
+is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead
+(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads
+(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs
+(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0
+t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead
+(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead
+(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t))
+(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d
+w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t))
+(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is
+v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat
+Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v
+(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/defs.ma".
+
+inductive sn3 (c: C): T \to Prop \def
+| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
+
+definition sns3:
+ C \to (TList \to Prop)
+\def
+ let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil
+\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))])
+in sns3.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sn3/defs.ma".
+
+include "LambdaDelta-1/pr3/props.ma".
+
+theorem sn3_gen_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
+(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
+T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))
+(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T
+y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0))))
+(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x))
+\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda
+(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
+(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda
+(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
+c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall
+(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c
+(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T
+t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1
+x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead
+(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall
+(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
+(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c
+(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2)
+\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4
+(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
+b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
+(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x
+x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
+T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T
+x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b)
+t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2)
+(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda
+(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b)
+x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4
+(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
+b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x
+x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
+T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T
+t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in
+(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0
+t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3
+(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c
+x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y
+H0))))) H))))).
+
+theorem sn3_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
+(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0:
+T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y:
+T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
+(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0:
+T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3
+c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T
+t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1:
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land
+(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
+(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1
+x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in
+(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead
+(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2:
+T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7:
+(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T
+(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let
+H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11
+\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0:
+Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7
+(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T
+(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2)
+(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c
+x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x
+t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x
+t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0)
+(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
+T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
+T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal
+T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f)
+x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c
+x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
+
+theorem sn3_gen_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
+(THead k u t)) \to (sn3 c u)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u:
+T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b:
+B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
+(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in
+(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
+c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f:
+F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
+(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in
+(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
+(sn3 c t)).H1)) H0)))))))) k).
+
+theorem sn3_gen_cflat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead
+c (Flat f) u) t) \to (sn3 c t)))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0:
+T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
+(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T
+t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
+(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2)
+\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2
+(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
+
+theorem sn3_gen_lift:
+ \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1
+(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1
+t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))
+(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq
+T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0)))))
+(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to
+(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1:
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to
+(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1
+c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d
+x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T
+t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0))
+\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d
+x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq
+T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2)))))
+H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T
+x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d
+t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let
+H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h
+d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to
+(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T
+x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2
+H4)))))))))))))) y H0)))) H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sn3/props.ma".
+
+include "LambdaDelta-1/drop1/fwd.ma".
+
+include "LambdaDelta-1/lift1/fwd.ma".
+
+theorem sns3_lifts1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to
+(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c
+(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda
+(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H)
+in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList
+ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c
+H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to
+(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0
+p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def
+(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
+(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1
+(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda
+(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t:
+TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1))
+(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sn3/defs.ma".
+
+include "LambdaDelta-1/nf2/dec.ma".
+
+include "LambdaDelta-1/nf2/pr3.ma".
+
+theorem sn3_nf2:
+ \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t
+(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P:
+Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2
+H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y)
+in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P:
+Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3
+(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
+
+theorem nf2_sn3:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c
+t u)) (\lambda (u: T).(nf2 c u)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda
+(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u))))
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let
+H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2
+c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u))
+(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c
+t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1
+x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1
+x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u:
+T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1
+u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x
+x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u))
+(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3))
+H2)))))) t H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sn3/nf2.ma".
+
+include "LambdaDelta-1/sn3/fwd.ma".
+
+include "LambdaDelta-1/nf2/iso.ma".
+
+include "LambdaDelta-1/pr3/iso.ma".
+
+theorem sn3_pr3_trans:
+ \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1
+t2) \to (sn3 c t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda
+(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2:
+T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
+(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3:
+T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3:
+(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let
+H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T
+t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let
+H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8
+\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P)))
+H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2
+H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P:
+Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
+
+theorem sn3_pr2_intro:
+ \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to
+(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1
+t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c
+t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to
+(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in
+((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall
+(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3
+c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0)))))
+(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall
+(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t
+t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5:
+T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3
+t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to
+((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7:
+((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4
+t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P:
+Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq
+T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10:
+(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to
+(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t:
+T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t
+t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t:
+T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3)
+\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5))
+H9))))))))))) t1 t2 H1 H3)) H2)))))))).
+
+theorem sn3_cast:
+ \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to
+(sn3 c (THead (Flat Cast) u t))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda
+(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0)))))
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2
+t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0:
+T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3:
+((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0
+t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1
+t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2:
+T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P:
+Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def
+(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c
+t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0
+x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12
+\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to
+(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T
+(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def
+(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1)
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14:
+(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat
+Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1
+H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1
+H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1)))
+(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1)
+((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1))
+(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3:
+T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall
+(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3:
+T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead
+(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c
+(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to
+(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14))))
+(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda
+(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda
+(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17
+\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead
+(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18
+\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16
+(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec
+t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to
+(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def
+(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat
+Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def
+(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0
+(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16:
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1
+H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0
+t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8)))
+H7))))))))) t H2)))))) u H))).
+
+theorem sn3_cflat:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u:
+T).(sn3 (CHead c (Flat f) u) t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f:
+F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1
+(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2
+(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
+
+theorem sn3_shift:
+ \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c
+(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
+H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
+(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
+v) t)).H2)) H0))))))).
+
+theorem sn3_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
+T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3
+(CHead c (Bind b) v2) t)))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
+(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda
+(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind
+b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3
+(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
+(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1
+(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
+(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
+v1)))))))))) t H0))))))).
+
+theorem sn3_gen_def:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
+i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
+(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
+i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
+Abbr c d v i H))))))).
+
+theorem sn3_cdelta:
+ \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
+(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
+C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
+\def
+ \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
+T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
+\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
+C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
+(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
+(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
+(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
+\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
+C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
+v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
+(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
+(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
+c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
+H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
+(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
+(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
+v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
+C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
+v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
+(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
+c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
+(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
+C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
+(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
+(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3
+(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
+(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
+(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
+H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
+t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
+(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
+in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda
+(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
+H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
+(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
+C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
+v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
+(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
+(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
+H0)))))).
+
+theorem sn3_cpr3_trans:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
+t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1)
+t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0))
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1)
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
+t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T
+t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1
+t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
+
+theorem sn3_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t:
+T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c
+u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t)
+t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_:
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
+t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c
+(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t:
+T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b)
+t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2:
+T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b)
+t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b)
+t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda
+(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda
+(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst)
+in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3)
+(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c
+(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b
+(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P:
+Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall
+(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
+b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let
+H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to
+(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3
+(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b
+(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
+\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to
+(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def
+(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0
+x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3
+(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P:
+Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind
+Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in
+(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P:
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let
+H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2)
+(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let
+H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in
+(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1
+\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda
+(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T
+(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P:
+Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0:
+T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0))))
+H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1
+t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst)
+t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P:
+Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20:
+(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1)
+in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P:
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let
+H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0:
+T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda
+(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans
+c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1
+H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20
+H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst
+t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b
+Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind
+b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
+(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b)
+t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq
+T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13:
+(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0:
+T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead
+(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
+T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in
+(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead
+(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0
+(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to
+(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda
+(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c
+(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def
+H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c
+(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r
+T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0))
+\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1
+(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T
+t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead
+(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20:
+(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0
+H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2
+\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18:
+(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c
+(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c
+(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind
+b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq
+T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1
+x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
+(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
+t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
+(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10)
+c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t
+H2)))))) u H)))).
+
+theorem sn3_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
+t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
+(Bind Abst) w t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
+(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
+c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
+Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
+y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst)
+w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead
+(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
+Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall
+(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w:
+T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w
+x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda
+(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
+c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x
+x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead
+(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
+(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c
+w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1:
+T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0:
+T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0
+x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1
+(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P)))
+\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in
+(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0
+x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to
+(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8:
+((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2
+t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3
+x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
+(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind
+Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead
+(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x
+(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst)
+t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq
+T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15:
+(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda
+(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to
+(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T
+(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def
+(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2))
+(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst)
+x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind
+T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
+(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
+x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
+(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
+H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c
+(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2
+x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl)
+x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0
+x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3
+(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0:
+T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def
+(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
+(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2
+x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl)
+t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let
+H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind
+T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2
+x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T
+x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
+(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def
+(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind
+Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall
+(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0
+H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead
+(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind
+Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
+H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
+(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind
+Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4
+(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in
+(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0)
+P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
+(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
+(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27:
+(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4)
+(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1
+x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
+\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x
+x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
+H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
+x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2
+c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
+Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2
+H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P:
+Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind
+(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl)
+x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def
+(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x
+(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))))
+(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4)
+((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead
+(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T
+x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
+Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4
+H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
+(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind
+Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4
+(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in
+(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0)
+P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
+(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
+(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25)))
+(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind
+Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr)
+x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
+\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x
+x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
+H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
+x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2
+c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
+Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23
+(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13)))))))
+H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead
+(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3
+x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0)
+\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T
+(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0]))
+(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0)
+\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in
+(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0
+H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x
+x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4))
+(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0:
+T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead
+(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in
+(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead
+(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0
+t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr)
+x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26:
+(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4)
+(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x
+x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0:
+T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def
+(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
+(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2
+c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4
+(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3)
+\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq
+T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P:
+Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
+(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr)
+x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def
+(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
+(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda
+(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30
+\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29
+(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15)
+(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3)))))
+H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14:
+(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq
+T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
+(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c
+(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
+Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
+H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
+x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
+(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _)
+\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in
+((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14)
+in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def
+(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0
+H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2
+H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b)
+x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b:
+B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3
+c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29
+\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_:
+False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5)
+x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12))
+H11))))))))) w H4))))))))))) y H0))))) H)))).
+
+theorem sn3_appl_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v:
+T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead
+(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef
+i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2:
+T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall
+(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i))
+t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
+(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1
+x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda
+(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P)))
+H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1)
+(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq
+T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
+Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda
+(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i)
+(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1
+x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T
+t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat
+Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P:
+Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c
+t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t
+(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c
+(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0)
+\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H
+x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))
+(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8:
+(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_:
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let
+H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i))
+t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r
+T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind
+T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c
+(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B
+T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead
+(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat
+Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2
+c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def
+(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to
+(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
+O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
+O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0)
+x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6))
+H5))))))))) v H0))))).
+
+theorem sn3_appl_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v
+(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
+(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
+(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead
+(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro
+T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3
+c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x:
+T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat
+Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
+t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
+Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
+i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift
+(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
+T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall
+(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead
+(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w))
+H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2
+(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat
+Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl)
+x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead
+(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8)
+in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
+(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c
+x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2
+(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P:
+Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat
+Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i
+H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u:
+T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
+T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16:
+(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead
+(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
+Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c
+(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x
+in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
+(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def
+(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead
+(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T
+(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0
+H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead
+(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x
+(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P:
+Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
+(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead
+(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0
+(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let
+H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22
+(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w))
+(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl)
+(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O
+w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda
+(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
+(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr)
+x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1
+(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0
+t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T
+(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20
+\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H
+(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2
+(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3)
+(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in
+((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d
+(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w)
+i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24
+\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20
+w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S
+i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0
+(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def
+H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
+(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c
+(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x
+x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w))
+(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat
+Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t]))
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
+w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to
+(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda
+(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
+w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3
+H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10:
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
+t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0
+x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c
+x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
+u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
+Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2
+x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t))
+(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
+x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2
+H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i)
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
+Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda
+(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
+x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t:
+T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in
+(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
+(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10))
+H9))))))))))))) y H1)))) H0))))))).
+
+theorem sn3_appl_cast:
+ \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v
+u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead
+(Flat Appl) v (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead
+(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3
+c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
+(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y:
+T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead
+(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
+(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v
+(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to
+(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat
+Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall
+(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0:
+T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead
+(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2
+(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x
+t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
+t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
+(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x
+t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0))
+(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t))))
+(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq
+T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat
+Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0
+(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
+x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda
+(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3
+c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1:
+T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T
+t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat
+Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
+x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0
+(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
+\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let
+H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to
+(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3:
+T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead
+(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3
+t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1
+(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
+\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
+(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda
+(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0
+x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat
+Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x
+(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast)
+x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq
+T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19:
+(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda
+(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to
+(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T
+(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def
+(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c
+x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2
+x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat
+Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1
+x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
+(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
+Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat
+Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x
+\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let
+H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
+x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5)))
+(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
+x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
+(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl)
+x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0)
+(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def
+(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat
+Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall
+(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3:
+T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead
+(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2
+(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))
+(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P)))
+H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18
+x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
+(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1)
+(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat
+Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead
+(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
+(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1)
+(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
+\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1)
+(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3:
+T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl)
+x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let
+H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in
+(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast)
+x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))
+(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda
+(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall
+(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat
+Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5
+(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29)))
+(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4))
+\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x
+x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead
+(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1)
+(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat
+Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl)
+x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
+(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl)
+x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1
+| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
+Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x
+x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32)
+in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat
+Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
+(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28
+x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x
+H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
+(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead
+(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl))
+x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat
+Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead
+(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P:
+Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x
+x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat
+Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2
+c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23)))))))
+H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat
+Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl)
+x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead
+(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead
+(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
+\Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1)
+(THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 |
+(TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
+Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x
+x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26)
+in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
+(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
+Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat
+Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
+(Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to
+(\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda
+(t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c
+(THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2
+H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1)
+(THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat
+Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3
+H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2)
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
+(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda
+(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x
+x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead
+(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13
+(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5)
+(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0
+x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2
+x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2
+H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat
+Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2)
+(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
+(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18:
+(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq
+T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda
+(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c
+(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P:
+Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5))
+H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6)
+x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast)
+x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4)
+H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O)
+O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5))))
+H4))))))))) y H0))))) H)))).
+
+theorem sn3_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
+T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u)
+(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v
+(THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
+(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0:
+T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O)
+O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0)))))))
+(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat
+Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2
+t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
+(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
+t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))))
+(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t
+(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c
+(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0:
+T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3
+c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b)
+t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat
+Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b)
+t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3)
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3
+(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2
+t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to
+(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
+x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1
+x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead
+(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0:
+T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3
+(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
+(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1
+(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0)
+H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
+t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
+(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0)
+H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda
+(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0))
+t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x
+(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b)
+t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
+t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0)
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
+b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3)
+(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0)
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c
+t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat
+Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead
+(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
+H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2)
+(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in
+(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind
+b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
+(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
+(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3
+x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1)
+x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl)
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P:
+Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3
+x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def
+(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3)
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3
+x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0:
+T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1
+(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27
+\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T
+t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4))))
+(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4)
+((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead
+(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead
+(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0
+H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b)
+t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
+Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32
+\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3
+c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x
+x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl)
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
+(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda
+(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c
+(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead
+(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead
+(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P:
+Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O
+H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
+(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0
+(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29))))
+(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat
+Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S
+O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P:
+Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in
+(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
+H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead
+(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b)
+t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r
+T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let
+H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead
+(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall
+(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def
+(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O
+H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b)
+t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S
+O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15))
+x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1)
+x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P:
+Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead
+(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26
+\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P))
+(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda
+(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead
+c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3
+(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2
+\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x
+x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl)
+(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T
+x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0:
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
+(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
+x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
+(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O
+H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
+(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
+H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P:
+Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in
+(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
+H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c
+(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda
+(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal
+T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift
+(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c
+c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26))))))
+H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift
+(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2)
+(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat
+Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans
+(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def
+(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
+(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
+O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1
+(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0:
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
+(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
+x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
+(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O
+H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
+(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
+H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx
+(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift
+(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c
+(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13))
+(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0:
+T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
+T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4)))))))
+(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1
+x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
+x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind
+b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
+(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10
+(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4)
+(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
+\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in
+((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14)
+in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def
+(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead
+c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda
+(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind
+Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def
+(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl)
+(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
+b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind
+b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0:
+B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to
+(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl)
+(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4
+(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5
+(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b
+(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
+\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind
+b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat
+Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def
+(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30
+\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_:
+False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20))
+H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b)
+t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
+Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1:
+B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T
+(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3
+(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
+(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead
+c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
+H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in
+(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
+(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
+\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
+((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
+(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def
+(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0
+H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1
+H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0)
+x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind
+b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead
+(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1
+(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def
+(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to
+(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
+O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5
+(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0:
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let
+H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq
+T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat
+Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def
+(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
+H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat
+Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat
+Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to
+(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda
+(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
+(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in
+(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0:
+Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2
+(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P))))))
+(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O
+x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c
+(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
+(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x)
+Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
+Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5)
+x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in
+(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def
+(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
+H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda
+(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def
+(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
+H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2
+c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
+(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x)
+(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind
+b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c
+(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat
+Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl)
+(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O
+x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O
+x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13))
+H12)))))))))))))) y H4))))) H3))))))) u H0))))).
+
+theorem sn3_appl_appl:
+ \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
+(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
+(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
+u1)))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
+(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T
+(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall
+(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
+(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
+y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to
+(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso
+y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2))))))
+\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t:
+T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2:
+T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
+(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x:
+T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2:
+T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
+(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
+(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
+(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T
+t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
+(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
+(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
+t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2
+(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c
+v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso
+t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2))))))
+\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5:
+((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0
+t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to
+(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2
+u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
+Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7:
+((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T
+t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead
+(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall
+(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to
+(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat
+Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T
+t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P:
+Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
+(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall
+(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
+(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3
+t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2
+(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P)))
+\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
+(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat
+Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x
+x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead
+(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def
+(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl)
+x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl)
+x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T
+t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18:
+(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t:
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
+Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat
+Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2
+H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c
+(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1
+x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat
+Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0
+x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0
+(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
+Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat
+Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
+(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26
+\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))
+((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P:
+Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda
+(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in
+((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27)
+in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t:
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
+x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let
+H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind
+T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))))
+(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
+(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))
+\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3
+(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c
+(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0
+x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0)))
+(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t:
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
+t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let
+H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind
+T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))
+(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36:
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17)
+(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda
+(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead
+(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1
+u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4
+H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat
+Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27
+(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4
+(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c
+t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c
+(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3
+x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0
+u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0)
+(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4)
+(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28))
+(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29
+(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x
+x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead
+(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat
+Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat
+Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
+(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23:
+(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda
+(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4
+x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl)
+t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
+Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind
+Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def
+(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl)
+x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P:
+Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
+t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
+(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall
+(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
+(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4)
+H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c
+(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead
+(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t:
+T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c
+t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso
+(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
+(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
+t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat
+Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing
+c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4))
+(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind
+Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind
+Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6
+(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5
+x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t:
+T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
+(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to
+P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1)
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort
+n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda
+(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T
+(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2)
+\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind
+Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5
+x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
+(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind
+Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow
+(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst)
+x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5
+x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
+| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
+(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 |
+(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4)
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
+\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3
+x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T
+t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr)
+x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_:
+T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5)
+(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind
+Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T
+(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40:
+(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def
+(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq
+T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k
+(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat
+Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5
+x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead
+(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind
+Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5
+x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22:
+(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4
+x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S
+O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4
+x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind
+T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
+(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8
+(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c
+(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T
+(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead
+(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P:
+Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
+t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
+(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall
+(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
+(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23)
+in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead
+(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4
+x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4:
+T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to
+(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
+(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
+Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
+t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat
+Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32
+(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c
+(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat
+Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead
+(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x)
+x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl
+x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
+(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift
+(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c
+(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3)
+x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x
+x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl)
+(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl
+(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind
+x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
+x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t:
+T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
+(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat
+Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow
+(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4
+x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl)
+(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T
+(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
+P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef
+i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T
+(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
+x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
+(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) |
+(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4)
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k
+v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let
+H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t)
+\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4
+x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4
+| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
+(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat
+Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0:
+K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0
+v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
+P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4
+(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq
+T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_:
+T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl)
+(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5
+t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43
+\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
+H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4
+(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))])
+in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5)))
+(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
+x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift
+(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead
+(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8
+(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat
+Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))))))))))
+x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T
+(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3
+(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_:
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let
+H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead
+(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3
+x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t))
+(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind
+Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17:
+(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T
+t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
+(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c
+(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
+Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
+H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
+x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x
+x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3)
+H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O)
+O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2
+H4))))))))) y H0))))) H))))).
+
+theorem sn3_appl_beta:
+ \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
+(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
+\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w
+t))))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
+T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
+Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind
+Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
+Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
+Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
+H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead
+(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind
+Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
+Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c
+(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl))))))))
+H1))))))))).
+
+theorem sn3_appl_appls:
+ \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads
+(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall
+(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
+(sn3 c (THead (Flat Appl) v2 u1))))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads
+(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead
+(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0:
+(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1
+(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads
+(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
+Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0
+H1))))))))).
+
+theorem sn3_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us:
+TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads
+(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H))
+(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3
+c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t)
+(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef
+i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil
+(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1
+in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
+TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref
+c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
+(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land
+(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2
+(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads
+(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3
+c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1)
+(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
+(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c
+t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
+(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda
+(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1)
+(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat
+Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl)
+(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9
+(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t
+u2))))))))) H5))) H3))))))) t0))) us)))).
+
+theorem sn3_appls_cast:
+ \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat
+Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3
+c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
+(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads
+(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u
+t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda
+(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0:
+TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads
+(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2))
+\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to
+(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to
+(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2)))
+\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u
+t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil
+u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c
+(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u:
+T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil
+u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
+(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1:
+T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat
+Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to
+(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall
+(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall
+(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c
+(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u
+t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl)
+(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1
+t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u
+t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
+(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead
+(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def
+(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3
+\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat
+Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
+(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c
+(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let
+H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in
+(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads
+(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1
+t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c
+(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in
+(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8
+(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2)
+(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl)
+(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P:
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons
+t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl)
+(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t
+Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
+
+theorem sn3_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
+T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind
+b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat
+Appl) vs (THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
+(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t:
+TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts
+(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0))))))
+(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u
+H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t:
+TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
+(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u
+t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl)
+(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0))))))))
+(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
+(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b)
+u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil)
+t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads
+(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead
+(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to
+(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u
+t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u)
+(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads
+(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1:
+T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
+v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
+(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
+(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3
+(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat
+Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
+t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda
+(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads
+(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c
+(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c
+(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2:
+T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
+t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind
+b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b
+H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v
+(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))
+(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0))
+t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead
+(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9
+Appl)))))))))) H4))))))))) vs0))) vs)))))).
+
+theorem sn3_appls_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
+(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c
+w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst)
+w t))))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us:
+TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead
+(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H:
+(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c
+w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0:
+TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0
+(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads
+(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3
+c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_:
+(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w:
+T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead
+(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads
+(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1:
+(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v
+t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead
+(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c
+w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl)
+v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat
+Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
+\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
+(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
+T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
+(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl)
+(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4:
+(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead
+(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3
+c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w
+t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat
+Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8
+\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
+(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
+t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
+t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
+
+theorem sn3_lift:
+ \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda
+(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d)
+\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d
+t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall
+(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c:
+C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c
+d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T
+(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i
+t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T
+(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3))
+(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda
+(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h
+i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T
+(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1
+x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T
+(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let
+H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10
+(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6)))))
+H5))))))))))))) t H))).
+
+theorem sn3_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d
+v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef
+i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let
+H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2)
+(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t:
+T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in
+(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i))
+(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0:
+C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda
+(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr)
+x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2
+(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S
+i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let
+H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H
+(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0
+(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in
+C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1)
+(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in
+((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d
+(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v)
+i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12
+\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v
+H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def
+(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d
+H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10))))
+H9))) t2 H6)))))) H4)) H3))))))))))).
+
+theorem sn3_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl)
+vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
+(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3
+c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O
+w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H))
+in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0:
+TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift
+(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_:
+(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat
+Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads
+(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1)))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat
+Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i)))))
+\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w))))
+\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef
+i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i)
+O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
+(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
+O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
+t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead
+(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4:
+(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0
+(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda
+(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i))
+u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to
+(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat
+Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2)
+(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2
+(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl))))))))
+H3)))))))) vs0))) vs)))))).
+
+theorem sns3_lifts:
+ \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h
+i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda
+(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t:
+TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0)
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c
+(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def
+H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
+(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj
+(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
+H4)))) H2)))))) ts)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/theory.ma".
+
+axiom pc3_gen_appls_sort_abst:
+ \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall
+(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u))
+\to False)))))
+.
+
+axiom pc3_gen_appls_lref_abst:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall
+(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to
+False))))))))
+.
+
+axiom pc3_gen_appls_lref_sort:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws:
+TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads
+(Flat Appl) ws (TSort n))) \to False))))))))
+.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/G/defs.ma".
+
+include "LambdaDelta-1/getl/defs.ma".
+
+inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def
+| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort
+(next g n))))
+| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w)
+\to (sty0 g c (TLRef i) (lift (S i) O w))))))))
+| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w)
+\to (sty0 g c (TLRef i) (lift (S i) O v))))))))
+| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
+T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead
+(Bind b) v t1) (THead (Bind b) v t2)))))))
+| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2:
+T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat
+Appl) v t2))))))
+| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1
+v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c
+(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty0/defs.ma".
+
+theorem sty0_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c
+(TSort n) x) \to (eq T x (TSort (next g n)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c
+t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda
+(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda
+(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_:
+C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1:
+nat).(eq T (TSort (next g n1)) (TSort (next g n)))) (refl_equal T (TSort
+(next g n))) n0 H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v:
+T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
+v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v
+(TSort n)) \to (eq T w (TSort (next g n)))))).(\lambda (H4: (eq T (TLRef i)
+(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
+(False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl
+i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g
+n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T
+(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (TSort n) H4) in (False_ind (eq T (lift (S i) O v)
+(TSort (next g n))) H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda
+(v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind
+b) v) t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g
+n)))))).(\lambda (H3: (eq T (THead (Bind b) v t1) (TSort n))).(let H4 \def
+(eq_ind T (THead (Bind b) v t1) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
+(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4))))))))))
+(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort
+(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let
+H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
+(False_ind (eq T (THead (Flat Appl) v t2) (TSort (next g n))) H4)))))))))
+(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
+v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 (TSort (next g
+n)))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g
+n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort n))).(let H6
+\def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
+(False_ind (eq T (THead (Flat Cast) v2 t2) (TSort (next g n))) H6))))))))))))
+c y x H0))) H))))).
+
+theorem sty0_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c
+(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq T x (lift (S n) O u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c
+t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T
+T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u
+t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O
+u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda
+(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C
+T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u
+t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n)
+O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0:
+nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort
+n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (TSort (next g n0)) (lift (S n)
+O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T
+(TSort (next g n0)) (lift (S n) O u))))))) H2))))) (\lambda (c0: C).(\lambda
+(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d
+(Bind Abbr) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_:
+(((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq T w (lift (S n) O
+u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 \def (f_equal T
+nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i]))
+(TLRef i) (TLRef n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
+nat).(getl n0 c0 (CHead d (Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n
+(\lambda (n0: nat).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T (lift (S n0) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n0) O w) (lift (S n) O
+u)))))))) (or_introl (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T (lift (S n) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O w) (lift (S n) O
+u)))))) (ex3_3_intro C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T (lift (S n) O w) (lift (S n) O t))))) d v w H6 H2 (refl_equal T
+(lift (S n) O w)))) i H5)))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind
+Abst) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T
+v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq T w (lift (S n) O u)))))))))).(\lambda (H4: (eq T
+(TLRef i) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in
+T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0)
+\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H4) in
+(let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind
+Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C T T
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v)
+(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T
+T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v)
+(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v)
+(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i
+H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e
+u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n)
+O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T
+(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v
+t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Bind b) v t2) (lift (S
+n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq T (THead (Bind b) v t2) (lift (S n) O u))))))) H4))))))))))
+(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T
+T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O
+t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2
+(lift (S n) O u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef
+n))).(let H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
+H3) in (False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O t)))))) (ex3_3 C T T
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Flat
+Appl) v t2) (lift (S n) O u))))))) H4))))))))) (\lambda (c0: C).(\lambda (v1:
+T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq T v2 (lift (S n) O u)))))))))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) v1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat
+Cast) v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat
+Cast) v2 t2) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u)))))))
+H6)))))))))))) c y x H0))) H))))).
+
+theorem sty0_gen_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
+T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda
+(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead
+(Bind b) u t2))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1)
+x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x))
+(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2))
+(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda
+(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g
+(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u
+t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
+(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind b) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g
+(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n))
+(THead (Bind b) u t2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
+v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v
+(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind b)
+u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda (H4:
+(eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2:
+T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O
+w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T
+v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind
+b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda
+(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2:
+T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O
+v) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (b0: B).(\lambda (c0:
+C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g
+(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u
+t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind
+b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda
+(H3: (eq T (THead (Bind b0) v t0) (THead (Bind b) u t1))).(let H4 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t0) (THead
+(Bind b) u t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _)
+\Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t0) (THead
+(Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead
+(Bind b) u t1) H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0
+b)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1))
+\to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u)
+t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in
+(let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t
+t2)) H1 t1 H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead
+(Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind
+b0) t) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u
+t3)))))) H9 u H7) in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead
+c0 (Bind b0) t) t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T
+(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T
+(THead (Bind b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0
+(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3:
+T).(sty0 g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3:
+T).(eq T t2 (THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B
+b0 (\lambda (b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in
+(eq_ind_r B b (\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0
+(Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead
+(Bind b) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
+u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u
+t3))) t2 H14 (refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5))
+H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u
+t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3))
+(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T
+(THead (Flat Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T
+(THead (Flat Appl) v t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+b) u t1) H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
+u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u
+t3)))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u
+t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2))
+(\lambda (t2: T).(eq T v2 (THead (Bind b) u t2))))))).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0
+(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
+u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda
+(H5: (eq T (THead (Flat Cast) v1 t0) (THead (Bind b) u t1))).(let H6 \def
+(eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex2 T (\lambda (t3:
+T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat
+Cast) v2 t2) (THead (Bind b) u t3)))) H6)))))))))))) c y x H0))) H))))))).
+
+theorem sty0_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x:
+T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g
+c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x:
+T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead
+(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T
+(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat
+Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
+u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
+t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n:
+nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H1) in
+(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
+(TSort (next g n)) (THead (Flat Appl) u t2)))) H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
+T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u
+t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5
+\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in
+(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
+(lift (S i) O w) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
+T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u
+t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5
+\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in
+(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
+(lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (b:
+B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0
+(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind
+b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u
+t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u
+t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T (\lambda (t3:
+T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v t2) (THead
+(Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda
+(t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 t2)).(\lambda (H2: (((eq
+T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3))
+(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))))))).(\lambda (H3: (eq T
+(THead (Flat Appl) v t0) (THead (Flat Appl) u t1))).(let H4 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t]))
+(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in
+(\lambda (H6: (eq T v u)).(let H7 \def (eq_ind T t0 (\lambda (t: T).((eq T t
+(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3))
+(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))))) H2 t1 H5) in (let H8
+\def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T
+u (\lambda (t: T).(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3:
+T).(eq T (THead (Flat Appl) t t2) (THead (Flat Appl) u t3))))) (ex_intro2 T
+(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl)
+u t2) (THead (Flat Appl) u t3))) t2 H8 (refl_equal T (THead (Flat Appl) u
+t2))) v H6))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl)
+u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
+v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to
+(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead
+(Flat Appl) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t1)
+H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3:
+T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Appl) u t3)))) H6))))))))))))
+c y x H0))) H)))))).
+
+theorem sty0_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall
+(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2:
+T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0
+g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2
+t2))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda
+(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T
+(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_:
+T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda
+(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0
+g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq
+T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_:
+T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2)))
+(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2)))))))))
+(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat
+Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast)
+v1 t1) H1) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g
+c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda
+(v2: T).(\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Cast) v2
+t2))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w:
+T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1
+t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g d v1 v2)))
+(\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda
+(t2: T).(eq T w (THead (Flat Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef
+i) (THead (Flat Cast) v1 t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda (v2:
+T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0
+g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O w) (THead
+(Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
+Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T
+v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_:
+T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2)))
+(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2
+t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5
+\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in
+(False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2)))
+(\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2:
+T).(\lambda (t2: T).(eq T (lift (S i) O v) (THead (Flat Cast) v2 t2)))))
+H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
+(\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind b) v) v1 v2)))
+(\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) t1 t3)))
+(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2
+t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1
+t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T T (\lambda
+(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3:
+T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind
+b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda (c0: C).(\lambda
+(v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
+(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda
+(t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead
+(Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v t0) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1)
+H3) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1
+v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2:
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Flat Cast) v2
+t3))))) H4))))))))) (\lambda (c0: C).(\lambda (v0: T).(\lambda (v2:
+T).(\lambda (H1: (sty0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat
+Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1
+v3))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v3:
+T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) v3 t2)))))))).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0
+(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
+T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3)))
+(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3
+t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) v0 t0) (THead (Flat Cast)
+v1 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0
+| (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast)
+v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast)
+v1 t1) H5) in (\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda
+(t: T).((eq T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3:
+T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0
+g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast)
+v3 t3))))))) H4 t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g
+c0 t t2)) H3 t1 H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t
+(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
+T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3)))
+(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2
+v1 H8) in (let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1
+H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3)))
+(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3:
+T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3
+t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2)))))))))
+H6)))))))))))) c y x H0))) H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty0/defs.ma".
+
+include "LambdaDelta-1/getl/drop.ma".
+
+theorem sty0_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
+e) \to (sty0 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c)
+\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda
+(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_:
+(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d
+(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g
+c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort
+(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v
+w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0:
+nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0
+w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3:
+(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0
+(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le
+i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0)
+in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h
+d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind
+nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
+(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
+H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
+Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S
+i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w)))
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
+d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T
+(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind
+nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i)
+(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S
+i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_:
+nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w))))
+(sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x
+(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i))
+w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
+(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
+i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
+(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
+h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat
+(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
+O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0
+(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
+c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h)
+(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
+h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
+h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
+i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
+T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h:
+nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift
+h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
+(H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h
+d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def
+(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d
+(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g
+c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0:
+C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h
+(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let
+H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
+(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0
+h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0
+(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h
+(minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S
+i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
+h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x
+d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i)
+O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g
+c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h
+(minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0
+(\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i))
+v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead
+x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S
+i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
+(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
+i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
+(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
+h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat
+(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
+O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0
+(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
+c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h)
+(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
+h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
+h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
+i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3
+t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h
+d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s
+(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v
+t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4))
+(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b)
+d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d)
+t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2
+b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift
+h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda
+(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g
+c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat
+Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v
+t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d)
+t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s
+(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl)
+d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h
+d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d
+(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d))))))))))))
+(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1
+v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d
+v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3
+t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4:
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s
+(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast)
+v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast)
+d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h
+(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0
+h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
+(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat
+Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast)
+v1 t3 h d))))))))))))))) e t1 t2 H))))).
+
+theorem sty0_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c
+t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2:
+T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda
+(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2))
+(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
+w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2
+in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g
+c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w
+x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i)
+O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0)))))
+H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
+T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g
+d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex
+T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda
+(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v)
+t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v
+i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v:
+T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b)
+v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v)
+t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0
+(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3)
+t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3
+x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead
+(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0:
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0
+t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2
+\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
+T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3:
+(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl)
+v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2))))))))
+(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
+v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T
+(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda
+(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat
+Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6
+\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
+T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7:
+(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast)
+v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7))))
+H6)))) H4))))))))))) c t1 t H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty1/props.ma".
+
+include "LambdaDelta-1/cnt/props.ma".
+
+theorem sty1_cnt:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c
+t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt
+t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3))))))
+(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0
+(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0
+(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n)))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0
+g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda
+(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d
+v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef
+i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v
+x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i)
+t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x
+H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T
+(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def
+H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2))
+(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2)))
+(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt
+x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2:
+T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v)
+(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1))
+(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i
+H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0:
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g
+(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g
+(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
+(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda
+(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2)
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead
+c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4:
+T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead
+(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v)))))
+H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4:
+T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
+(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4))
+(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda
+(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda
+(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v
+t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v
+t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0:
+C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1
+v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2:
+T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2
+t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4:
+T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2
+t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead
+(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda
+(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0
+t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g
+c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat
+Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2)
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1
+x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0
+x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4))
+(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat
+Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty0/defs.ma".
+
+inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def
+| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2))
+| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c
+t t2) \to (sty1 g c t1 t2)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/sty1/defs.ma".
+
+include "LambdaDelta-1/sty0/props.ma".
+
+theorem sty1_trans:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c
+t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g
+c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g
+c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g
+c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0
+g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
+
+theorem sty1_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
+T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead
+(Bind b) v t1) (THead (Bind b) v t2))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1
+t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead
+(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g
+(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead
+(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_:
+(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v
+t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c
+(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t)
+H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))).
+
+theorem sty1_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
+(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat
+Appl) v t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1
+g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3:
+T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1)
+(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t:
+T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v
+t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t
+t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1
+(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))).
+
+theorem sty1_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
+e) \to (sty1 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h
+d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1
+t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop
+h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0
+c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1:
+((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to
+(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2:
+(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3)
+(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
+
+theorem sty1_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c
+t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2:
+T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1
+t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1
+t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2:
+T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))).
+
+theorem sty1_abbr:
+ \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w)
+\to (sty1 g c (TLRef i) (lift (S i) O w)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w:
+T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c
+(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v
+t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2
+H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c
+(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t
+t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2)
+(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w
+H0)))))))).
+
+theorem sty1_cast2:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c
+t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1:
+T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c
+v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat
+Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1)
+(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast
+g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1
+t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to
+(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead
+(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda
+(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g
+c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1
+v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast)
+v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1
+g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def
+(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4:
+T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3:
+T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda
+(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g
+c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat
+Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast
+g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/defs.ma".
+
+definition subst:
+ nat \to (T \to (T \to T))
+\def
+ let rec subst (d: nat) (v: T) (t: T) on t: T \def (match t with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k
+u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))]) in subst.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst/defs.ma".
+
+theorem subst_sort:
+ \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort
+k)) (TSort k))))
+\def
+ \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort
+k)))).
+
+theorem subst_lref_lt:
+ \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T
+(subst d v (TLRef i)) (TLRef i)))))
+\def
+ \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i
+d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i)))
+(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))).
+
+theorem subst_lref_eq:
+ \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v)))
+\def
+ \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq
+T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with
+[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift
+i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))).
+
+theorem subst_lref_gt:
+ \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T
+(subst d v (TLRef i)) (TLRef (pred i))))))
+\def
+ \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d
+i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef
+(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred
+i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d)
+(le_bge d i (lt_le_weak d i H)))))).
+
+theorem subst_head:
+ \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d:
+nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w
+t)))))))
+\def
+ \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d:
+nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst/fwd.ma".
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+theorem subst_lift_SO:
+ \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S
+O) d t)) t)))
+\def
+ \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq
+T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d:
+nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n)))
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T
+(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n))
+(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq
+T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n
+d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n)))
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T
+(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d
+(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T
+(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n)))
+(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0))
+(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T
+t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0)
+(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
+(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef
+(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O)
+(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d
+(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0))
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v
+(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O)
+d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k
+t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v
+(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1)))
+(f_equal3 K T T T THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v
+(lift (S O) (s k d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))) (subst d v
+(THead k (lift (S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S
+O) d t0) (lift (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1))
+(lift_head k t0 t1 (S O) d)))))))) t)).
+
+theorem subst_subst0:
+ \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0
+d v t1 t2) \to (eq T (subst d v t1) (subst d v t2))))))
+\def
+ \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3))))))
+(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t:
+T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i)
+(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T
+(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0
+t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t))
+(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0)))
+(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0
+i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i)))
+(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1
+u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t:
+T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t))
+(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k
+(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst
+i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0:
+T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s
+k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t)))
+(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i))
+(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k:
+K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i:
+nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k
+i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst
+i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u
+t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda
+(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T
+(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead
+k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0
+u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u
+t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u
+t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0
+u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i)
+v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s
+k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r
+T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T
+(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0
+u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i
+v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda
+(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k
+i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4)))
+(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4))
+(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1
+t3 i))))))))))))) d v t1 t2 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+theorem dnf_dec2:
+ \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S
+O) d v))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda
+(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d:
+nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n)
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d
+v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort
+n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T
+(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n:
+nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T
+(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v:
+T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d
+(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind
+nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0
+w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift
+(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w
+(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S
+O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n)
+(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w)
+(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n
+(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d
+H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n)
+(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred
+n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
+(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
+t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda
+(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d)
+in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0
+d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1)
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O)
+d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0
+(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
+(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S
+O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))))
+(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift
+(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d
+v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w
+t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
+T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w)
+in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift
+(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S
+O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s
+k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda
+(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w
+(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d
+w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0
+t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2))
+(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6)
+(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5))))))
+(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T
+(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex
+T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x))
+(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def
+H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T
+(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d
+v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d)
+x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O)
+(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d)
+x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1
+H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex
+T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x:
+T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in
+(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s
+k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S
+O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
+k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1)
+(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v:
+T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O)
+d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
+(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2
+t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T
+(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v))))
+(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T
+(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda
+(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda
+(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d)
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1)
+(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift
+(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1)
+t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift
+(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3))
+(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T
+(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x)
+(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
+k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
+T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v))))
+(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k
+d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d
+x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift
+(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O)
+d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
+
+theorem dnf_dec:
+ \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or
+(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
+\def
+ \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t
+d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v:
+T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S
+O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t
+(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v:
+T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1
+\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T
+(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d
+v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d
+x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t
+(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t
+(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t
+(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex
+T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d
+v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T
+(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0
+(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v:
+T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x)
+(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d
+x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d
+x)))) t H1))) H0)) H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/defs.ma".
+
+inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
+| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift
+(S i) O v)))
+| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i:
+nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v
+(THead k u1 t) (THead k u2 t))))))))
+| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1:
+T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i
+v (THead k u t1) (THead k u t2))))))))
+| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i:
+nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2:
+T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2
+t2)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+theorem subst0_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
+i v (TSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n)
+(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda
+(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda
+(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_:
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def
+(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2)))))
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n))
+\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t)
+(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
+(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda
+(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0
+i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k:
+K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k
+u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))).
+
+theorem subst0_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
+i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0
+i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v))))
+(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0:
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n))
+\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0:
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0:
+nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj
+(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n)
+(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
+(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T
+(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2
+t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0:
+T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0
+(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s
+k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T
+(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u
+t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
+(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef
+n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda
+(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2
+t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
+
+theorem subst0_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
+(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1)
+x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda
+(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
+(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
+t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda
+(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k
+u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda
+(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1
+t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1
+t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef
+i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S
+i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T
+(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0)
+v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0:
+T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq
+T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
+u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0:
+K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _)
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0)
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T
+u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
+(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t)
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda
+(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead
+k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind
+T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda
+(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
+(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0
+(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7)
+in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1
+H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
+(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1
+t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1)
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T
+(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0
+H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2:
+T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0
+t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
+t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1
+t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda
+(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead
+k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u
+u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T
+(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2)
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0
+(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
+t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s
+k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1:
+K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2
+(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T
+(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s
+k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1
+t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s
+k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
+(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2)
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T
+(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2)
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3:
+T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0)
+v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5))
+H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda
+(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4:
+(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
+t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _)
+\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T
+u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t:
+T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t
+t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1
+(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1)))
+(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T
+t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in
+(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12
+k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T
+(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
+(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda
+(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
+i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0
+(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda
+(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0
+v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
+i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0
+H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))).
+
+theorem subst0_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d
+u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n:
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n))
+x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t:
+T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d))))
+in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n)
+t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S
+(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2:
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
+n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
+(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
+(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq
+T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
+(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
+(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))))
+(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0)
+O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T
+(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O
+(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3)))
+(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
+n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
+T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
+(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
+h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
+(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
+u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n))
+H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2:
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
+n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4))))
+(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
+(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift
+h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0
+t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t
+t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0))
+(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i
+d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d))))
+in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus
+i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d))
+t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t)
+t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
+d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S
+(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h
+d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
+t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k
+(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S
+(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h
+(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
+T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d)))
+t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t)
+x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T
+x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T
+(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
+(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda
+(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d)))
+t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
+t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h
+(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k
+(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1
+t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3:
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2)))
+(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d)))
+t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i
+d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
+(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
+T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda
+(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0)
+x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2
+T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3:
+T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i
+d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S
+(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus
+i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0))
+H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
+(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
+(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d))
+x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s
+k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h
+(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i
+u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T
+(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1))
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
+t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda
+(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind
+T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
+(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i
+d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d))
+(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i
+d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d))))
+(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0
+H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
+(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d
+u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
+(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda
+(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x
+(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d))
+t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
+d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
+(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda
+(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i
+d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d))
+(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7
+(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
+(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda
+(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k
+i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T
+(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S
+(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S
+(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
+x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
+t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S
+(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind
+nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k
+(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus
+i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S
+(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2:
+T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead
+k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus
+i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T
+(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k
+x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u
+t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
+(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d)))
+(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0
+i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
+(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i
+H2))))))))))))) t1)).
+
+theorem subst0_gen_lift_false:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u
+(lift h d t) x) \to (\forall (P: Prop).P)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x:
+T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i
+(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
+Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda
+(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n))
+x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda
+(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in
+(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u:
+T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i:
+nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1:
+(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P
+(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda
+(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind
+(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda
+(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0:
+nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n
+H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n))
+(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d
+H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P
+(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n
+h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d
+h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h
+n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall
+(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h))
+\to ((subst0 i u (lift h d t0) x) \to (\forall (P:
+Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall
+(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to
+((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P:
+Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus
+d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P:
+Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2:
+T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1))
+(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k
+u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)))
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2:
+T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
+(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2:
+T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u
+(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda
+(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7:
+(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda
+(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
+(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2:
+T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
+(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k
+(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1)
+x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h))
+(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i
+(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda
+(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7:
+(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d)
+t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d
+t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
+
+theorem subst0_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h)
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h
+d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n:
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus
+d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0
+i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2
+T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i
+h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d
+(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
+n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
+(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
+(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq
+T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
+(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
+\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus
+d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
+(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
+T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
+(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S
+(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
+h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
+h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S
+(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2))))
+(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S
+(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n)
+t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h
+d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u)
+(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n
+h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
+u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
+nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
+(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n
+h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d
+(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n
+d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i
+H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h)
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0:
+((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
+nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0
+t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda
+(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0))
+(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d)
+t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x
+(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t)
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda
+(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
+k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
+(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0))))
+(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2:
+T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u
+(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
+(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u
+(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
+(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0
+(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda
+(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7:
+(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r
+T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
+(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u
+(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
+(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
+h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u
+(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0)))
+(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k
+d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4))
+(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2)))
+(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T
+(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0
+(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
+(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6:
+(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t)
+x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3)))
+(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T
+(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0
+(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d
+t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0)
+t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda
+(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1)
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2)
+(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
+(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t
+t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n:
+nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r
+d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1))
+(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))
+(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0
+(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t
+x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h))
+(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h)
+(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
+k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift
+h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u
+(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0)
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq
+T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0)
+t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda
+(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T
+(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
+k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d)
+x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda
+(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t
+t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda
+(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t
+x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T
+(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead
+k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda
+(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d
+(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d
+t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12
+\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9
+(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T
+(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal
+T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2
+H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d))
+x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind
+nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i
+H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u
+(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/fwd.ma".
+
+theorem subst0_refl:
+ \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d:
+nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort
+n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda
+(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef
+n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O
+u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O
+u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1
+P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P:
+Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u
+t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0
+d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead
+k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T
+(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1)))
+(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T
+(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _)
+\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def
+(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6
+P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1)
+(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T
+(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2:
+T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k
+t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2)
+\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def
+(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s
+k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0
+t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0
+x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1
+x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead
+_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2)
+\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T
+t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1
+t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u
+t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0
+t1 (THead k t0 t1) d H1)))))))))) t)).
+
+theorem subst0_lift_lt:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i
+(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall
+(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d
+t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda
+(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t:
+T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v))))
+(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0)))
+(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O
+v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t:
+T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S
+i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d
+v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r
+(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda
+(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_:
+(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall
+(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d
+u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt
+i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d)
+t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d
+(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t))
+(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d
+u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h
+d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k
+u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h
+d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1:
+((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0)
+(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda
+(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3
+\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s
+k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v)
+(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r
+T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0
+(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead
+k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h
+(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind
+nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v)
+(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s
+k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s
+k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d
+u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0))
+(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h
+d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d:
+nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v)
+(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda
+(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d:
+nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d
+(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda
+(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0))
+(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0:
+nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0
+t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1)
+(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t
+(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k
+d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift
+h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v)
+(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k
+d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s
+k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d)
+(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d
+(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0))
+(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
+
+theorem subst0_lift_ge:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
+(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0
+(plus i h) u (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n:
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le
+d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v:
+T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T
+(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift
+(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0
+(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda
+(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r
+nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0))
+O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S
+i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d
+(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d
+H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le
+d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t:
+T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T
+(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0
+h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift
+h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1)
+(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0
+h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2
+t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda
+(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d:
+nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d
+t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3
+\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0:
+nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1
+(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0)
+(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead
+k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda
+(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t))
+(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k
+d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k
+u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d)))))))))))))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda
+(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to
+(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k:
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0
+t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k
+i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le
+d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n:
+nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0)
+(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T
+(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0
+h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift
+h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1)
+(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0
+h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d
+i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead
+k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
+
+theorem subst0_lift_ge_S:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d
+t1) (lift (S O) d t2))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat
+(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d
+t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O))
+(plus_sym i (S O)))))))))).
+
+theorem subst0_lift_ge_s:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s
+(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda
+(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/props.ma".
+
+theorem subst0_subst0:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
+u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
+T).(subst0 (S (plus i j)) u t t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
+T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3)))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i))
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
+(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t:
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t
+(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v
+u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i))
+(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t:
+T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0:
+nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i
+u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda
+(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0
+i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1
+x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0:
+T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i))
+u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst
+u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k:
+K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i:
+nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1:
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
+i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0:
+T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
+i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x:
+T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0
+(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n:
+nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in
+(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n
+u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T
+(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S
+(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3
+u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0
+H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3:
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda
+(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t
+u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
+(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u:
+T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0
+(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t
+t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4:
+(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t:
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
+t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0
+x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda
+(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))
+(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t:
+T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0
+u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0
+(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def
+(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9
+(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t:
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
+t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5)
+(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4)))))
+(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))).
+
+theorem subst0_subst0_back:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
+u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
+T).(subst0 (S (plus i j)) u t2 t)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
+T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4)))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i))
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
+(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t:
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift
+(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0
+(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0
+(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda
+(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1:
+((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to
+(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus
+i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3:
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v
+u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0:
+T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3
+(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t)
+t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0
+(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k
+u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0))
+(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0
+i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v:
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0
+(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall
+(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1
+t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda
+(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2:
+(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t:
+T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0
+(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3
+x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def
+(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4
+(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k
+(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i)))
+(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u
+t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead
+k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6
+u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1:
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k:
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0
+t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3:
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v
+u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t:
+T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3
+(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3)
+t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6:
+(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i
+u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda
+(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0
+i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1
+x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r
+nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus
+i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0
+i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k
+(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0)
+t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0
+x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i))
+H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2
+t1 t2 H))))).
+
+theorem subst0_trans:
+ \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0
+i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1
+t3)))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to
+(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3:
+T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false
+v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0
+(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1:
+T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1:
+((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda
+(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k
+u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t)))
+(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3
+(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda
+(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3:
+T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3
+t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3)
+(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0
+i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead
+k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda
+(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4:
+T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k
+u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k
+u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda
+(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0:
+T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3
+H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T
+t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5:
+(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T
+(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0))
+(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3))
+(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0:
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0
+(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0
+t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4:
+T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda
+(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))
+(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s
+k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2)))
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0
+(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2
+t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq
+T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0
+(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x
+t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda
+(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0)
+t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u
+t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5:
+T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))
+(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4
+(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k
+u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3
+i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0
+t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3)
+t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0
+x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3)
+t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3))
+(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda
+(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1
+u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0
+u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2:
+(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0)
+v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4:
+(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda
+(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1
+t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3)))
+(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead
+k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda
+(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0
+i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4
+H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5)))
+(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5:
+T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4
+(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead
+k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1
+u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda
+(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda
+(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0
+i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda
+(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t:
+T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0
+x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0
+H4))))))))))))))) i v t1 t2 H))))).
+
+theorem subst0_confluence_neq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
+nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
+(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
+(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t))))))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
+nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n:
+nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4:
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq
+nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5:
+T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda
+(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2
+(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq
+T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v)
+t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda
+(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n:
+nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda
+(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda
+(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in
+False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift
+(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with [])
+in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v:
+T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0
+i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2:
+nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t:
+T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda
+(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2:
+nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq
+nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t)))
+(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2
+(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T
+(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4:
+T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead
+k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T
+(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0
+u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4:
+T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t)
+t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq
+T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k
+x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t)
+t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3:
+T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda
+(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
+k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda
+(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t)
+(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2
+H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead
+k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda
+(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t
+t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0
+x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x)
+(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4))
+(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0
+i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3))
+(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k))
+t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq
+T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0
+u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t
+t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4
+t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3:
+T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0
+x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4:
+T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4))))
+(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i
+v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8:
+(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda
+(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
+k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7)
+(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4))
+(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda
+(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0:
+(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2:
+T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2))
+\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k
+i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3)
+t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq
+T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda
+(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t:
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda
+(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k
+u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0
+i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u
+x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t:
+T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2
+k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T
+(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2)
+u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda
+(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2
+t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t))
+(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7:
+(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2
+T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i
+v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u)
+(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t:
+T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq
+nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k
+i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s
+k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7:
+(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4))
+(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k
+u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda
+(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k
+u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u
+x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5))
+(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k
+i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t:
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0
+x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda
+(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k
+x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0)))))
+(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t))
+(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to
+False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_:
+(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj
+k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k
+i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3
+t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2:
+T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1:
+((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2)
+\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t))
+(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3:
+((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4)
+\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3
+t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4:
+T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k
+u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4:
+T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T
+(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2)
+u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4
+t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda
+(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k
+u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0
+i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x
+t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2
+u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x
+t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0
+i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2)
+u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda
+(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3
+t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda
+(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x
+x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3
+i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0:
+T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x
+x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11))))))
+(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i
+i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4:
+T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_:
+T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2)
+u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4
+(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4)))
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda
+(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0
+x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3
+t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v
+x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2
+u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t:
+T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T
+(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v
+(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3
+x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0
+x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both
+v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t:
+T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq
+nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k
+i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat
+(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda
+(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2
+H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2
+H4)))))))))))))))))) i1 u1 t0 t1 H))))).
+
+theorem subst0_confluence_eq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2)
+(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t)))
+(subst0 i u t1 t2) (subst0 i u t2 t1))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to
+(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5:
+T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda
+(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef
+i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T
+(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v)
+t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2)
+(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda
+(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2)
+(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t:
+T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2
+(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v
+t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2:
+T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0
+i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda
+(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T
+t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t
+t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2)
+(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
+(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3
+t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq
+T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead
+k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
+(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x
+t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda
+(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v
+(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v
+(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x)
+(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x
+t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead
+k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k
+x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2
+x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t))
+(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4:
+T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x
+t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x
+t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t)
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead
+k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3))
+(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0
+v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t)
+(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3))
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t)
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0:
+T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x
+x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x
+t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x
+t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t)
+t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t)
+(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6))
+(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x
+t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x
+t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t
+k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t)
+(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3))
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t)
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x
+i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3:
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+k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead
+k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2
+i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5))
+(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))).
+
+theorem subst0_confluence_lift:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i
+t2)) \to (eq T t1 t2)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0
+i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i
+t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t:
+T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S
+O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2)
+(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def
+(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i
+H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t))
+(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t:
+T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O)
+i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i
+t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1)
+x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S
+O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O))
+(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift
+(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1)
+(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n))
+(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2)))
+(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i
+t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i)
+(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O)
+i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2)))
+(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+include "LambdaDelta-1/lift/tlt.ma".
+
+theorem subst0_weight_le:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift
+(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O
+v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd
+g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g
+H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
+u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1)
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2))
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i)))
+\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S
+(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f))))))))))))))))
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0)
+(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g
+O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map
+f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b))
+(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g
+m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g
+u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0)
+(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2
+H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall
+(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f
+m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le
+(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda
+(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s
+k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map
+f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b:
+B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s
+(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
+(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f
+t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2))
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f
+g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1)))
+(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1))
+(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5
+(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v))
+(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O
+(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5:
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b))
+(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
+(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g
+t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5:
+(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2)
+(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus
+(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1
+f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
+
+theorem subst0_weight_lt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v:
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
+(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd
+f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n
+(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O
+O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
+u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1)
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2))
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i)))
+\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f))))))))))))))))
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f
+u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd
+g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
+(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
+(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
+f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1))
+(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
+(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
+(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt
+(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map
+g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1:
+T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
+k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v
+t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2)
+(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
+(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2))
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
+t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1
+f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2)))
+(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S
+(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2)
+(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O
+v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f
+u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i)
+f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v
+t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m)
+(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m)
+(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O)
+m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g
+i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v
+(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m))))
+\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2)
+(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
+(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
+f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
+(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
+(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t
+z H))))).
+
+theorem subst0_tlt_head:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt
+(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
+(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map
+(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n
+(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m)))
+(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n:
+nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda
+(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_:
+nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u)
+(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda
+(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda
+(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
+
+theorem subst0_tlt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z
+(THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx
+(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def
+| subst1_refl: subst1 i v t1 t1
+| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst1/defs.ma".
+
+include "LambdaDelta-1/subst0/props.ma".
+
+theorem subst1_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
+i v (TSort n) x) \to (eq T x (TSort n))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T
+t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0
+i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x
+H))))).
+
+theorem subst1_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
+i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift
+(S n) O v))))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or
+(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl
+(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O
+v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v
+(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq
+nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i)
+(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x
+H))))).
+
+theorem subst1_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
+(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst1 (s k i) v t1 t2))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1)
+x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1
+t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1
+t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal
+T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda
+(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2
+(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda
+(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1
+x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1
+x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3:
+T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0:
+T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v
+t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1)
+(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda
+(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1
+i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0
+x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4)))))))
+H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
+
+theorem subst1_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst1 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S
+(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1)
+(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2:
+T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1))
+(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u)
+(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h
+(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1
+t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d))
+x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T
+t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1
+(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x
+H))))))).
+
+theorem subst1_gen_lift_eq:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u
+(lift h d t) x) \to (eq T x (lift h d t))))))))))
+\def
+ \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d
+h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t)
+(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda
+(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t
+u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
+
+theorem subst1_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h)
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst1 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1)
+x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2:
+T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
+h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1
+(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2:
+T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3:
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3))
+(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1
+(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d
+x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3:
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0
+H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h
+d H1 H0)))) x H)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst1/defs.ma".
+
+include "LambdaDelta-1/subst0/props.ma".
+
+theorem subst1_head:
+ \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1
+i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s
+k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2))))))))))
+\def
+ \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k:
+K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i
+v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k
+i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t)))
+(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k
+i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k
+v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
+t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1
+(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead
+k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1)
+(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k
+i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both
+v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
+
+theorem subst1_lift_lt:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1
+i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i
+(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d:
+nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u)
+(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i
+d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d
+t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d:
+nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h
+(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d
+H1 h))))))) t2 H))))).
+
+theorem subst1_lift_ge:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
+(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1
+(plus i h) u (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t:
+T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h
+d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u
+(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda
+(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1)
+(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
+
+theorem subst1_ex:
+ \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2:
+T).(subst1 d u t1 (lift (S O) d t2))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex
+T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n:
+nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n)
+(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d
+u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n))
+(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d
+(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda
+(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O)
+d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n)
+t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S
+O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T
+(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T
+(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u)
+(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n)
+t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S
+O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d
+H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n)
+(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t:
+T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef
+(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t:
+T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift
+(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T
+(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d:
+nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2:
+T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u
+(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u
+t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
+(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex
+T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda
+(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d)
+x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d
+t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k
+d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t
+(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k
+x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
+
+theorem subst1_lift_S:
+ \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i
+(TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
+\def
+ \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h:
+nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i
+t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_:
+(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift
+(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef
+h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n))
+(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S
+i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H:
+(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n))
+(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n)
+(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T
+(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i
+(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0))
+(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0))))
+(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0:
+nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef
+h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T
+(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n))))
+(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef
+n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h)
+(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n
+(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0:
+nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O
+(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n
+(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n))
+(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n))
+(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h)
+(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n))
+(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt
+n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T
+(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i
+(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i
+(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n
+(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S
+(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i)
+H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i:
+nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t)
+(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i:
+nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i)
+t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1:
+(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
+t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0))))
+(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1:
+T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
+t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i
+h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S
+(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift
+(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k
+(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i))
+(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst1/fwd.ma".
+
+include "LambdaDelta-1/subst0/subst0.ma".
+
+theorem subst1_subst1:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
+u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
+T).(subst1 (S (plus i j)) u t t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
+t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
+(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
+(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl
+(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1
+t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1
+i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S
+(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1
+y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0:
+T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3)))))
+(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda
+(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
+t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t:
+T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0)
+(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3:
+(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0
+(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t:
+T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T
+(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
+t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7:
+(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1
+t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1
+x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j
+H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))).
+
+theorem subst1_subst1_back:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
+u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
+T).(subst1 (S (plus i j)) u t2 t)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t
+t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
+(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
+(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl
+(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1
+t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1
+i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0:
+T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0))))
+(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S
+(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i
+j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T
+(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u
+t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S
+(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1
+x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t:
+T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x
+(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4)))))
+(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))).
+
+theorem subst1_trans:
+ \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1
+i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1
+t3)))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3:
+T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda
+(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1
+t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3
+(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0:
+T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans
+t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))).
+
+theorem subst1_confluence_neq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
+nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
+(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
+(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t))))))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
+nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t:
+T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2)
+\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3))
+(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not
+(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda
+(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2:
+T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not
+(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2
+(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4:
+T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1
+u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1
+i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda
+(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x
+(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4)))))
+(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2
+H2))))) t3 H1)))))))) t1 H))))).
+
+theorem subst1_confluence_eq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
+i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t:
+T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t)))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2:
+T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3))
+(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0:
+(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda
+(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2:
+T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i
+u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1
+i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t:
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u
+t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u
+t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t))
+(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4)
+(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)))
+(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda
+(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2
+T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2
+(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T
+(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i
+u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i
+u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5:
+(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda
+(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4
+x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t:
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u
+t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2
+t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1
+i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4)))
+(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))).
+
+theorem subst1_confluence_lift:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
+i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i
+t2)) \to (eq T t1 t2)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1)
+(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1
+i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0:
+(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i
+t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1
+t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda
+(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda
+(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4
+\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u
+(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
+(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O)))
+H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1:
+(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3:
+T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2
+(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T
+(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1
+t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0
+(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6:
+(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t:
+T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in
+(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r
+nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i
+(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6:
+(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def
+(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in
+(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5)))
+H3))))))) y H0))) H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst0/tlt.ma".
+
+include "LambdaDelta-1/subst/props.ma".
+
+include "LambdaDelta-1/sty1/cnt.ma".
+
+include "LambdaDelta-1/ex0/props.ma".
+
+include "LambdaDelta-1/wcpr0/fwd.ma".
+
+include "LambdaDelta-1/pr3/wcpr0.ma".
+
+include "LambdaDelta-1/ex2/props.ma".
+
+include "LambdaDelta-1/ex1/props.ma".
+
+include "LambdaDelta-1/ty3/sty0.ma".
+
+include "LambdaDelta-1/csubt/csuba.ma".
+
+include "LambdaDelta-1/ty3/fwd_nf2.ma".
+
+include "LambdaDelta-1/ty3/nf2.ma".
+
+include "LambdaDelta-1/wf3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+inductive TList: Set \def
+| TNil: TList
+| TCons: T \to (TList \to TList).
+
+definition THeads:
+ K \to (TList \to (T \to T))
+\def
+ let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t:
+T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u
+(THeads k ul t))])) in THeads.
+
+definition TApp:
+ TList \to (T \to TList)
+\def
+ let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts
+with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t
+(TApp ts0 v))])) in TApp.
+
+definition tslen:
+ TList \to nat
+\def
+ let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O
+| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen.
+
+definition tslt:
+ TList \to (TList \to Prop)
+\def
+ \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/tlist/defs.ma".
+
+theorem tslt_wf__q_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
+\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
+\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
+ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
+(tslen ts)))))).
+
+theorem tslt_wf_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
+TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
+TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
+Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
+(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
+TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
+nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
+(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
+m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
+\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
+(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
+H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
+ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
+
+theorem theads_tapp:
+ \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T
+(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads
+k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda
+(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k
+v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0
+(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads
+k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
+
+theorem tcons_tapp_ex:
+ \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
+(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
+\def
+ \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
+TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
+ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
+ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
+(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
+TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
+T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
+(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
+t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
+ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
+(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
+TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
+(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
+(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
+nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
+TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
+t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
+(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
+nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
+t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
+nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
+(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
+(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
+(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
+
+theorem tlist_ind_rev:
+ \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
+TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
+TList).(P ts))))
+\def
+ \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
+((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
+t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
+(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
+TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
+TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
+\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
+\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
+(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
+T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
+(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
+t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
+(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
+H4))))) H3))))))) ts2)) ts)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+definition wadd:
+ ((nat \to nat)) \to (nat \to (nat \to nat))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n
+with [O \Rightarrow w | (S m) \Rightarrow (f m)]))).
+
+definition weight_map:
+ ((nat \to nat)) \to (T \to nat)
+\def
+ let rec weight_map (f: ((nat \to nat))) (t: T) on t: nat \def (match t with
+[(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0)
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr
+\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f
+u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f
+O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O)
+t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f
+t0)))])]) in weight_map.
+
+definition weight:
+ T \to nat
+\def
+ weight_map (\lambda (_: nat).O).
+
+definition tlt:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/tlt/defs.ma".
+
+theorem wadd_le:
+ \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
+nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to
+(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
+((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
+nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
+nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le
+(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
+
+theorem wadd_lt:
+ \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
+nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to
+(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
+((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
+nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
+nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0))
+(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0)))
+n))))))).
+
+theorem wadd_O:
+ \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_:
+nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat
+(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n).
+
+theorem weight_le:
+ \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t)
+(weight_map g t)))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall
+(n0: nat).(le (f n0) (g n0))))).(le_n (weight_map g (TSort n))))))) (\lambda
+(n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
+(H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) (\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1:
+T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1))))))
+\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1))
+(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0:
+B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0)
+(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0)
+(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus
+(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus
+(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr
+\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g
+t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g
+O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O)
+t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g
+t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
+(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus
+(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1))
+(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S
+(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g
+H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0)))
+(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0))
+(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n))))))))))))
+(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f
+t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
+(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
+(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1))
+(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map
+f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1)
+(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O
+(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g
+t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
+(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0)
+(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0)
+(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f
+O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n))))))))))))
+b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n))))
+\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g
+t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus
+(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g
+t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1)
+(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t).
+
+theorem weight_eq:
+ \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f
+t) (weight_map g t)))))
+\def
+ \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym
+(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n:
+nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n)
+(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0:
+nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))).
+
+theorem weight_add_O:
+ \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t)
+(weight_map (\lambda (_: nat).O) t))
+\def
+ \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_:
+nat).O) (\lambda (n: nat).(wadd_O n))).
+
+theorem weight_add_S:
+ \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O)
+O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
+\def
+ \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O)
+(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_:
+nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m
+(le_O_n m)) n)))).
+
+theorem tlt_trans:
+ \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to
+(tlt u t)))))
+\def
+ \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u)
+(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u)
+(weight v) (weight t) H H0))))).
+
+theorem tlt_head_sx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead
+k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
+(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
+(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
+(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_:
+nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_:
+nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map
+(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map
+(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O)
+u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda
+(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O)
+t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S
+(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u)
+(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_:
+nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k).
+
+theorem tlt_head_dx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt
+(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead
+k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
+(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
+(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S
+(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_:
+nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_:
+nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S
+(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u)
+(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O)
+u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
+(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd
+(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t
+(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t)
+(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t)))))))
+(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_:
+nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus
+(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_:
+nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_:
+nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map
+(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t)
+(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map
+(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_:
+nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S
+(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u)
+(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_:
+nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda
+(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u:
+T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))
+(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_:
+nat).O) t)))))) k).
+
+theorem tlt_wf__q_ind:
+ \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to
+Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0
+t))))) P n))) \to (\forall (t: T).(P t)))
+\def
+ let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
+T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t)
+n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight
+t)))))).
+
+theorem tlt_wf_ind:
+ \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t)
+\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t)))
+\def
+ let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
+T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
+Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v)
+(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind
+(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0:
+T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat
+(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
+(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P
+t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt
+(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight
+v))))))))))))) t)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/pr3_props.ma".
+
+include "LambdaDelta-1/arity/pr3.ma".
+
+include "LambdaDelta-1/asucc/fwd.ma".
+
+theorem ty3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity
+g c t2 (asucc g a1))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g
+c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity
+g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
+a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda
+(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1)))
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3
+(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_:
+(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g
+a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10:
+(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0:
+T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1))))
+(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3
+x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
+g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g
+(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12
+g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11)))))
+H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A
+(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0
+(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort
+g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g
+d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1))
+(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g
+c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr
+c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d
+t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind
+A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
+a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A
+(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0
+(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g
+d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g
+d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7))))
+H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity
+g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b:
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b)
+u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1))
+(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
+g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b)
+u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u
+x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A
+(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1:
+A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1:
+A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead
+(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead
+c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc
+g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A
+(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0
+(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4)
+(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g
+x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to
+((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1:
+A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0
+(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0
+(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4
+(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u
+t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1)))
+x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr
+not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g
+(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind
+Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead
+(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4)
+(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5
+(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead
+x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1)
+H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead
+x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3
+x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g
+x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1))
+(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0
+(arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t3 x0
+H12) (arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t4
+(asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda
+(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
+(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0
+u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v
+(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1))
+(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
+g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda
+(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def
+H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity
+g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
+c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda
+(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t)
+(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in
+(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1
+a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
+(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))
+(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda
+(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g
+a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0)
+(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13:
+(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0)
+(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in
+(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A
+x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1
+x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2
+(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17)
+in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1
+x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1
+(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc
+g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc
+g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10)))))
+H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1:
+A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
+a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A
+(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1))
+(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
+g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
+Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3
+x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A
+(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
+a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1))
+(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1))))
+(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0
+t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat
+Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4)
+(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g
+x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0
+(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7)))))
+H4)))))))))) c t1 t2 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/arity.ma".
+
+include "LambdaDelta-1/sc3/arity.ma".
+
+theorem ty3_predicative:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u:
+T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P:
+Prop).P)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u:
+T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u
+v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v
+x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def
+(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def
+(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind
+A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1:
+A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c
+(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8
+\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda
+(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g
+x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def
+(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9)
+in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead
+x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g
+Abst c v t u H1)))))))))).
+
+theorem ty3_repellent:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1:
+T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead
+c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P:
+Prop).P)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1:
+T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2:
+T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O
+u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0:
+T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x:
+T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3
+\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop
+(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3
+(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2
+t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O)
+O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5
+(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind
+Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda
+(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g
+(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1:
+A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity
+g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def
+(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in
+(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda
+(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity
+g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g
+x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g
+c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1)
+(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2)
+H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w)
+t (lift (S O) O u2) H0))))))))))).
+
+theorem ty3_acyclic:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def
+(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in
+(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
+(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g
+c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x
+(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
+
+theorem ty3_sn3:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in
+(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u
+(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t
+x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t
+x H1))))) H0))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pc3/dec.ma".
+
+include "LambdaDelta-1/getl/flt.ma".
+
+include "LambdaDelta-1/getl/dec.ma".
+
+theorem ty3_inference:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2:
+T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0:
+C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2:
+T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind
+(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or
+(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4)
+\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall
+(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_:
+((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
+False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)))
+(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda
+(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n)))))
+(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3
+c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
+T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let
+H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d)
+\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n)
+t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1:
+(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead
+e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g
+c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)))
+(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2
+(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2))
+in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g
+x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T
+(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3:
+T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g
+x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2
+(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr)
+x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall
+(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3:
+T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x
+H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T
+(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef
+n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))
+(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2
+(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2
+(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))
+(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2
+(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O
+t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind
+Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0
+(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4)
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10))
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat
+_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0
+(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind False
+H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind
+Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0
+(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4)
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10))
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
+ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat
+_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0
+(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind False
+H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda
+(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T
+(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef
+n) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n)
+t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False
+(\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift
+(S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr)
+x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind
+x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4)
+(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in
+(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0
+(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0
+(Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2)
+n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2
+| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind
+Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr)
+x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let
+H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t)))
+H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5))
+H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead
+c0 (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda
+(c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H19)))))))) H12))
+H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7:
+(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst)
+x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind
+x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4)
+(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in
+(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0
+(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0
+(Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2)
+n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda
+(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2
+| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind
+Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst)
+x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let
+H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t)))
+H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5))
+H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S
+n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0:
+C).(getl n c2 (CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def
+(eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5
+H20))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5))))))
+H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P:
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (\lambda (t3:
+T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))) False (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr)
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5
+False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_:
+(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst)
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5
+False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1
+t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
+T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g
+c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0:
+T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to
+(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3
+t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3)))
+(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall
+(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
+False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3:
+T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1
+t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall
+(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
+False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in
+(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2
+t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0)
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False)))
+(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda
+(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b)
+t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to
+False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2
+(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T
+(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3
+g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g
+c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t
+t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2
+(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t)
+t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda
+(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3
+g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3
+g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b
+t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b)
+t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead
+(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3)
+\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0)
+t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b)
+t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4:
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda
+(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7
+x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4:
+((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g
+c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_:
+T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3
+g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0
+t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead
+(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead
+c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5))))))
+H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt
+c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
+t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda
+(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t
+t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3
+g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead
+(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3)
+\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3
+c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
+t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3
+c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3:
+T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T
+(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)))
+(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0
+(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
+c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3:
+T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0:
+T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)))
+(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3:
+T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
+False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def
+(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def
+H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u))
+(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall
+(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3:
+T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def
+(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0
+x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u:
+T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u:
+T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_:
+T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2
+v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or
+(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3
+c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2
+(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3
+v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3:
+T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3
+g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda
+(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23
+\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24
+\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1))
+H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl)
+t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
+False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))
+(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred
+g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0
+x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0
+(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3:
+T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3
+g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda
+(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda
+(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5))
+t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22:
+(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0
+\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5
+(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5)
+H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3
+(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3
+H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2
+t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to
+False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
+False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0)
+t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat
+Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3
+g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2
+t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead
+(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0
+(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind
+Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5))
+(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex
+T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3:
+T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T
+(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind
+Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind
+Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t
+(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst)
+x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t
+t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt
+c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1
+t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def
+(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3:
+T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T
+(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
+(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0
+(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
+c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3:
+T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0:
+T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
+(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2
+x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0
+t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0)
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
+(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast)
+t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat
+Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t
+x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to
+False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0)
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))
+(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0)
+t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3))
+(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False
+(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda
+(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def
+(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15
+x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g
+c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda
+(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3:
+T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda
+(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0
+t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_:
+(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0
+t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3
+H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to
+False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to
+False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0)
+t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3))
+(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False
+(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda
+(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4:
+T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5
+x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6))))))
+H4))) f H2))) k H1))))))) t2))) c t1))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/G/defs.ma".
+
+include "LambdaDelta-1/pc3/defs.ma".
+
+inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def
+| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t)
+\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to
+(ty3 g c u t2))))))))
+| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next
+g m))))
+| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u:
+T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to
+(ty3 g c (TLRef n) (lift (S n) O t))))))))
+| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u:
+T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to
+(ty3 g c (TLRef n) (lift (S n) O u))))))))
+| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to
+(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b)
+u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))
+| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to
+(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3
+g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u
+t)))))))))
+| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2)
+\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1)
+(THead (Flat Cast) t0 t2))))))).
+
+inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
+| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c
+TNil u)))
+| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts:
+TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/props.ma".
+
+include "LambdaDelta-1/pc3/fsubst0.ma".
+
+include "LambdaDelta-1/getl/getl.ma".
+
+theorem ty3_fsubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
+t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind
+Abbr) u)) \to (ty3 g c2 t2 t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
+(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3:
+T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind
+Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2:
+T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2
+c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2
+t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u
+t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2:
+C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c
+(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c
+t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
+\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u
+t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
+u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6)
+e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda
+(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2
+t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u
+(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3
+(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6:
+(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c
+c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr)
+u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8)
+t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c
+t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4
+H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i:
+nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0
+i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0:
+C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to
+(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i
+u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr)
+u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m))))))))
+(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda
+(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3:
+T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_:
+(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind
+Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g
+m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda
+(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i:
+nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2
+t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
+t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda
+(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c
+(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead
+e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3:
+T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5:
+(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S
+n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda
+(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4:
+T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0:
+nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C
+(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind
+Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0)
+H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
+c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d
+(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal
+C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u)
+(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e
+(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C
+e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let
+H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4)))
+H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift
+(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14))
+u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda
+(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5:
+(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift
+(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c
+c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind
+Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b)
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0))
+(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0
+H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n)
+(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1
+(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda
+(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow
+d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind
+x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
+(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14)
+in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind
+x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n
+c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus
+i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind
+Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i
+(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n
+H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6))
+in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd
+(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind
+Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda
+(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3
+(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1
+x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
+c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def
+(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d
+(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3:
+T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r
+C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let
+H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u)))
+H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0:
+nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0)))
+(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
+c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n
+n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr
+g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n))
+u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n
+(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr)
+x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13))
+H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0))
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
+(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11:
+(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0
+x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
+c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def
+(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15
+\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d
+(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr
+x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3:
+T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C
+x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let
+H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4)))
+H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0:
+nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0)))
+(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
+c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n
+n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr
+g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S
+n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S
+n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S
+(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))
+H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u
+(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1)))))))
+(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3:
+C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c
+(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0))
+(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq
+T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3
+g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0:
+nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def
+(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11
+\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0
+(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e
+(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono
+c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13
+\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
+n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15
+\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11
+d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d
+(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4:
+T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3
+(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n)
+(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d
+(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i
+n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
+Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2:
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind
+Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0:
+T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n)
+c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3:
+T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3
+(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n)
+t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr)
+u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S
+n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O
+u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n)
+O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
+(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
+(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
+Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S
+n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n
+H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e:
+C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3
+(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def
+(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind
+(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S
+n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3
+d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n)
+(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11:
+(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
+t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in
+(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
+(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14)
+in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind
+x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n
+c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus
+i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind
+Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i
+(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n
+H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6))
+in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3
+O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst
+g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n))
+u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus
+i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u
+(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n)))
+(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u
+u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq
+C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda
+(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind
+x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda
+(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
+Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst)
+u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let
+H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3)))
+H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i
+(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
+B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind
+nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e
+(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3
+(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead
+x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n)))
+(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0
+x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back
+(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e
+(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus
+i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S
+n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1
+(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda
+(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S
+n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in
+((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in
+((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3]))
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B
+Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda
+(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def
+(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d
+H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2
+(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n)
+(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr)
+u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n
+i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20
+(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in
+(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2
+(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e
+(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
+d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
+(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4
+n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus
+i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e
+(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
+d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
+(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop
+Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n))
+u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e
+(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
+d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
+(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8))
+H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c
+c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda
+(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0
+c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
+u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift
+(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O
+u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n)
+O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
+(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0:
+nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind
+Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
+(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
+Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S
+n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n
+H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda
+(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i:
+nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2
+t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
+t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2:
+(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall
+(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u)
+t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind
+Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0:
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead
+(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
+\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5:
+(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i
+c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead
+(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6:
+T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
+u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c
+t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5
+(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
+(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i
+u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8:
+(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T
+(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3)))
+(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c
+(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10:
+(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead
+c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3))
+(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g
+c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0
+H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0
+c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b)
+x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x)
+(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind
+Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3)
+(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd
+i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3
+(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0
+(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead
+c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
+H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead
+(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
+t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda
+(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3))
+(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9:
+(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda
+(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g
+(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u
+t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3
+x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x
+(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3
+(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b)
+u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5
+H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T
+t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
+u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c
+t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq
+T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda
+(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
+(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
+T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead
+(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3
+x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c
+(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_:
+(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3)
+(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1)
+(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c
+u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S
+i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind
+b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))
+(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead
+(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead
+(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b)))
+e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead
+c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
+H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5))))))
+(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda
+(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g
+(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b)
+u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3
+x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2
+t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind
+b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g
+(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2
+(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u)
+(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind
+Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead
+(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c
+c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2)))
+(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead
+(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead
+(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind
+b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
+T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))
+(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5
+(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead
+(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind
+T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead
+(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g
+(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3
+(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3))
+(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T
+(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind
+b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead
+c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u
+x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3
+x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i
+u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead
+c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3
+(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b)
+i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3)
+(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind
+b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3)
+(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead
+c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i)
+u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u
+x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))))))
+(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3
+(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0
+(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0
+H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9))))
+H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6)))
+(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6:
+T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
+u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9:
+(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2
+x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead
+(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3
+t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0:
+T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1
+i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3
+(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3
+(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3
+(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b)
+u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8))
+(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3
+t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq
+T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda
+(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
+T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind
+b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6))
+(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2:
+T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6:
+T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
+(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind
+b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x)
+(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x
+H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i
+u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0
+(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1
+H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c
+(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u
+t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u
+t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e
+H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead
+c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head
+(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3
+(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i)
+u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2
+(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind
+b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head
+(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c:
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1:
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
+Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3:
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
+Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i:
+nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4:
+(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead
+(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c
+(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind
+Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat
+Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead
+(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2:
+T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead
+(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl)
+x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v)
+(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0))))
+(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c
+(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0)
+x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind
+Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda
+(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c
+(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
+x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind
+Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead
+(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda
+(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w
+(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1))
+(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst
+t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u
+t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0
+H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
+(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u
+t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead
+(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x
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+g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7))
+(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda
+(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e
+(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3
+H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda
+(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda
+(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl
+i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5
+(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T
+(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6:
+T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat
+Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8:
+(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2:
+T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat
+Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat
+Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x
+t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2)
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda
+(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat
+Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g
+c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0
+(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x
+t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u
+c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2
+(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3
+(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i
+u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat
+Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3))
+i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3)
+(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e
+H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e
+H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead
+(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2
+t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6)))
+(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead
+(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast)
+t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead
+(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3)))
+(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0
+(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda
+(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast)
+u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_:
+T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_:
+T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead
+(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5
+(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11:
+(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda
+(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat
+Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3
+(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i
+u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1)
+(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u
+c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1
+(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3
+t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0
+(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c
+(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat
+Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead
+(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3
+i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst
+i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2
+t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
+
+theorem ty3_csubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1
+(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g
+c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2:
+C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1
+(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
+
+theorem ty3_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1
+t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e
+(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2
+t)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
+(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2
+H1) e H0))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/defs.ma".
+
+include "LambdaDelta-1/pc3/props.ma".
+
+theorem ty3_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
+(TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t
+x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda
+(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u:
+T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1
+t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e:
+T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9
+\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in
+(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2
+H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
+(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in
+T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _)
+\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in
+(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g
+n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d
+(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_:
+(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T
+(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t))
+H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t:
+T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d
+(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5
+\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0
+(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort
+(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let
+H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
+(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6)))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w
+u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n))
+u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
+Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n))
+(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v)
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w
+(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
+(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort
+(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort
+n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
+H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2))
+H6))))))))))) c y x H0))) H))))).
+
+theorem ty3_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
+(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t
+x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0)))))))
+(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1)
+t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e
+u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3
+c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u
+t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u
+t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6:
+(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n)
+H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift
+(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7)
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n)
+H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13
+H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0)
+t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0
+(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst)
+x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0
+(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10))))))))))))))))
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef
+n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in
+(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next
+g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t:
+T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S
+n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4:
+(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 |
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef
+n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d
+(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O
+t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O
+u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O
+t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5))))))))))))
+(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3
+g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S
+n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5
+\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
+with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _)
+\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0
+(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in
+(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda
+(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0
+(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O
+t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0
+(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
+n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to
+(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead
+c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind
+b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5:
+(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind
+b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda
+(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead
+(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t:
+T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind
+Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6
+\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in
+(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u
+t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u
+t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat
+Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t)
+(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))) H6))))))))))) c y x H0))) H))))).
+
+theorem ty3_gen_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
+T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq
+T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2
+T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0:
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda
+(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3:
+T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind
+b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u
+t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0:
+T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
+c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
+t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind
+b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u
+t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind
+b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4
+(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3
+g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T
+(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_:
+T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3
+g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0)
+t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b)
+u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4)))
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1
+(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13))))))
+H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
+(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m)))))
+(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0
+t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind
+b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1)
+H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0
+u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst)
+u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d
+(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1
+t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def
+(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind
+(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2)
+(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
+H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1:
+(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T
+T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda
+(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3)
+t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4)))
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind
+b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind
+b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return
+(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0
+| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0
+t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
+(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0)
+u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10:
+(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead
+(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead
+c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5:
+T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_:
+T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in
+(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3
+t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
+(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda
+(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_:
+T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10)
+in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0)
+t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2))))
+(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0
+(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4)
+t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind
+b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3:
+T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T
+u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_:
+T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0
+(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3:
+T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
+(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))
+(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
+t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))
+t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10))))))))
+H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0:
+T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u
+t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0
+t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0
+t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5:
+(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T
+(THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3
+c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t)))))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2
+t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
+(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_:
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2
+t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind
+(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
+(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t)))
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))
+H6))))))))))) c y x H0))) H))))))).
+
+theorem ty3_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead
+(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T
+(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind
+Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y:
+T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u
+t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u:
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0))
+t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat
+Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
+T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u
+(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u
+(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq
+T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda
+(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda
+(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7)
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead
+(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w
+v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
+T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0:
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
+t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13:
+(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0:
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w
+(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10))))))))))))))))
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat
+Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w
+v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda
+(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind
+Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u
+(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0:
+T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead
+(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
+(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O
+t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0
+t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5)))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind
+Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w
+u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5
+\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in
+(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0:
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
+(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0:
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1
+t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w
+(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g
+(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq
+T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u
+t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0
+t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6)))))))))))))
+(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0
+u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0
+t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0
+t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u
+t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
+T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead
+(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _)
+\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0)
+(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 |
+(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let
+H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0:
+T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T
+v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let
+H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v
+(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
+u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0
+u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0:
+T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1))
+(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda
+(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda
+(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl)
+w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v
+(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
+u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10
+H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat
+Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t:
+T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_:
+T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
+t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
+(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
+(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
+True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))
+H6))))))))))) c y x H0))) H)))))).
+
+theorem ty3_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall
+(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0:
+T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2))
+(\lambda (t0: T).(ty3 g c t2 t0))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T
+(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3
+T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3
+g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0:
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0
+(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast)
+t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t))
+(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2
+t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
+t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
+(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1
+t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3
+t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal
+T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def
+(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T
+(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3
+g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2
+t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3
+(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat
+Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4
+t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))
+(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_:
+T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0:
+T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3
+g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4:
+T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2)
+H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def
+(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in
+(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort
+(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
+t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3
+T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3
+g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef
+n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0
+(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1
+t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u
+t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
+(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2))
+(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead
+(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
+(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead
+(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
+Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
+t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat
+Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead
+(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T
+(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T
+(THead (Bind b) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
+Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda
+(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
+Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
+t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
+(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3
+T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u
+t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
+t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2
+t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
+(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T
+(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w
+(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0:
+T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
+Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g
+c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4:
+(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0
+(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0)
+(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0)
+(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
+t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9
+\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to
+(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_:
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let
+H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11
+\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to
+(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_:
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let
+H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r
+T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5
+t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
+(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T
+t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
+Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
+c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g
+c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat
+Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2))
+(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2))
+H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
+
+theorem tys3_gen_nil:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T
+(\lambda (u0: T).(ty3 g c u u0))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil
+u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_:
+TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda
+(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq
+TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0:
+T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList
+TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda
+(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts:
+TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to
+(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t
+ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee:
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind
+(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
+
+theorem tys3_gen_cons:
+ \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall
+(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts)
+(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u)
+(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind
+g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to
+(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1:
+T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
+return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
+\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0)
+(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1:
+(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0
+u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0)
+(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t
+ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList
+return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _)
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal
+TList TList (\lambda (e: TList).(match e in TList return (\lambda (_:
+TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1]))
+(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def
+(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land
+(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList
+ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind
+T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3
+g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/arity_props.ma".
+
+include "LambdaDelta-1/pc3/nf2.ma".
+
+include "LambdaDelta-1/nf2/fwd.ma".
+
+theorem ty3_gen_appl_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
+x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g
+c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in
+(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0
+x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl)
+w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))
+(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda
+(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def
+(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t:
+T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead
+(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t:
+T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3
+g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t)))))
+(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c
+(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind
+Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c
+(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c
+(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def
+(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c
+x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
+b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10
+(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13
+(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda
+(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u:
+T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c
+(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead
+(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6))
+(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w
+Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead
+(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind
+Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5
+x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2
+(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3))))))))
+(ty3_gen_appl g c w v x H))))))).
+
+theorem ty3_inv_lref_nf2_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c
+(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to
+((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda
+(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t
+u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c
+u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda
+(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2:
+T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift
+(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2
+c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to
+((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0
+t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda
+(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10
+\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11
+\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to
+(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0:
+T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y
+\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2
+H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq
+T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2:
+T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m))
+u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in
+(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5)))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7:
+(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
+O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
+n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))))))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7:
+(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
+O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
+n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0
+(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y
+d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2
+(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq
+T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i)
+O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0:
+T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda
+(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S
+i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
+i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u)
+t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0
+(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
+u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda
+(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0
+u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda
+(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
+(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2)
+\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
+u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
+(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to
+(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq
+T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl)
+w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead
+(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0:
+T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
+t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0
+t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda
+(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to
+((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast)
+t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0
+(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2
+t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T
+u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
+
+theorem ty3_inv_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c
+(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0:
+T).(eq T u (lift (S i) O u0))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1:
+(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
+
+theorem ty3_inv_appls_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1:
+T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to
+((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S
+i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u))
+u1))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t:
+TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t
+(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u:
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t
+(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H:
+(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c
+u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in
+(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u:
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1)))
+(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def
+(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r
+T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i)
+O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda
+(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u)
+(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2))))))))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall
+(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef
+i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
+(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u))
+u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c
+(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t
+(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T
+T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind
+Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_:
+T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst)
+u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u:
+T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u)))
+u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat
+Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t
+x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def
+(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
+x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3
+c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1)))
+(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0)
+x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def
+(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c
+(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i)
+O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O
+u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
+(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O
+x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead
+(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
+(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S
+i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c
+(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t
+Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
+
+theorem ty3_inv_lref_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c
+(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i
+j)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda
+(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda
+(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0
+H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift
+(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S
+i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
+in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x
+(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
+(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
+j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
+(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S
+i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i)
+j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4)))))
+H2))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/arity.ma".
+
+include "LambdaDelta-1/pc3/nf2.ma".
+
+include "LambdaDelta-1/nf2/arity.ma".
+
+definition ty3_nf2_inv_abst_premise:
+ C \to (T \to (T \to Prop))
+\def
+ \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi:
+T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs:
+TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w
+u)) \to False)))))))).
+
+theorem ty3_nf2_inv_abst_premise_csort:
+ \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise
+(CSort m) w u)))
+\def
+ \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda
+(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst)
+wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs
+(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d
+(Bind Abst) wi) H False))))))))).
+
+theorem ty3_nf2_inv_all:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
+t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat
+(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H)
+in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
+(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w)
+u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2:
+(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g
+c t x H2 H0)))) H1)))))))).
+
+theorem ty3_nf2_inv_sort:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t
+(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort
+n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda
+(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def
+(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2
+T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u))))
+(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u:
+T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))
+(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
+m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
+i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w)
+u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind
+Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w:
+T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda
+(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0
+x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0)
+x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H
+(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
+(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead
+(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n))))
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
+Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7:
+(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0
+x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst
+c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat
+(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n:
+nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c
+x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n:
+nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n)))
+(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
+m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
+i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def
+(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in
+(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T
+t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x)
+(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort
+n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda
+(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x)
+(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
+i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda
+(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x))
+(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g
+c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n:
+nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
+TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef
+x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat
+Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1))
+(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
+(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda
+(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n:
+nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda
+(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
+x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))
+x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3)))))))
+H2)) H1)))))))).
+
+theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux:
+ \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise
+c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead
+(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to
+(ty3_nf2_inv_abst_premise c w2 u2))))))))
+\def
+ \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d:
+C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi))
+\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi))
+(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2:
+T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind
+Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi:
+T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst)
+wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift
+(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t
+(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t
+(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl)
+vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst)
+w1 u1) H0))))))))))))))).
+
+theorem ty3_nf2_inv_abst:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u:
+T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to
+((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_:
+T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g
+c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v
+u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w)
+v))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u:
+T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c
+t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w
+u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in
+(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq
+T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c
+w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))))
+(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0:
+T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2
+c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0)
+u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind
+Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0:
+T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t
+(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2
+(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3
+g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r
+T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst)
+w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
+(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T
+(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x)))
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w
+x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T
+(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead
+(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0)))
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2)))
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
+(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4)
+(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g
+(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b:
+B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w
+v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c
+x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind
+b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def
+(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w
+H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst)
+t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w
+x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w))
+H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c
+x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9))))
+(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t
+H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort
+n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let
+H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H
+(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c
+(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq
+T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
+g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v
+u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t
+H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t
+(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
+(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
+(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0
+(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef
+x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind
+Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads
+(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u
+(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind
+Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind
+Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w
+(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1))
+(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T
+T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1))
+(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v))))))))
+(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c
+(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to
+((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
+(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2:
+T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda
+(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
+Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2
+T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x
+v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O
+t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
+Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef
+x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda
+(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O
+x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind
+Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
+Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda
+(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c
+(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0)
+(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda
+(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda
+(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15:
+(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let
+H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
+Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13))
+(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0:
+T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2:
+T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to
+((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
+(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2:
+T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13:
+(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda
+(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind
+Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl)
+t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_:
+T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))
+(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1
+(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let
+H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let
+H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1))
+(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
+Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads
+(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c
+x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_:
+(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T
+(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
+Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1))
+(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil
+(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
+g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
+x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))
+H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads
+(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3
+(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
+g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
+x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
+v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1))
+(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2
+(THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons
+t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0:
+T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind
+Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
+v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat
+Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t
+H5))))))) H4)) H3))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/ty3.ma".
+
+include "LambdaDelta-1/ty3/subst1.ma".
+
+include "LambdaDelta-1/ty3/fsubst0.ma".
+
+include "LambdaDelta-1/pc3/pc1.ma".
+
+include "LambdaDelta-1/pc3/wcpr0.ma".
+
+include "LambdaDelta-1/pc1/props.ma".
+
+theorem ty3_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
+t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2)
+\to (ty3 g c2 t2 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
+(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to
+(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0:
+T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c
+c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u:
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2:
+C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4
+t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0
+c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2
+H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2
+H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2:
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m)
+t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m))))
+(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
+Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2:
+((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g
+c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2:
+T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3:
+T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda
+(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2
+(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5:
+(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda
+(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)))))))
+(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n
+H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0:
+T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2)
+\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2:
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n)
+t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u)))
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind
+Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind
+Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g
+c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u
+(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S
+n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n)
+O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1
+H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n
+H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda
+(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to
+(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b:
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b)
+u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2)
+\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2:
+C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead
+(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5:
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u
+t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with
+[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u
+t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda
+(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8:
+(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda
+(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u
+(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u
+(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead
+(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow
+(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq
+T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
+\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead
+(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead
+(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in
+T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind
+b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T
+t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to
+(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind
+T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to
+((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))
+(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead
+(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead
+(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind
+T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to
+(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda
+(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3
+t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x:
+T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7:
+T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6)
+(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind
+b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x)
+(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2
+t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3
+(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17))
+(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u
+u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b)
+u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b))
+t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b)
+u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4
+H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind
+b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7)
+\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6)
+t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2)
+H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to
+((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) |
+(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq
+T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u
+t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
+b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2)
+H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2)
+\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7
+H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T
+(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead
+(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
+\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5)
+(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind
+Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B
+(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
+(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u
+t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to
+((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to
+((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda
+(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T
+(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O
+u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq
+T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4)
+\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4
+(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w)
+t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to
+((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u
+t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda
+(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0:
+B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7:
+T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def
+(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr
+H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7))
+(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x:
+T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda
+(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr)
+u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead
+c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead
+(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22)
+(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1
+c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20
+(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18)
+c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2
+t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind
+Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2
+(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18)))))
+(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u)
+(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4
+H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11))
+H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq
+T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda
+(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8)
+\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in
+lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
+t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
+t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7)
+\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u
+t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0
+| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5))
+(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 |
+(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in
+(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2)
+\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4
+(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda
+(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b
+Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda
+(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_:
+T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4
+(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda
+(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind
+b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5
+t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0
+(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8
+t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda
+(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in
+(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind
+b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead
+c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4)
+t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B
+Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda
+(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def
+(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O
+(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2
+(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4)
+(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S
+O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4
+(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25:
+(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u
+t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind
+T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O
+(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat
+(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S
+O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3)
+(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3
+x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead
+(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1
+u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2
+(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u
+(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u))))))))))))
+H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2
+(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S
+O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return
+(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in
+H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead
+c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift
+(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u)
+(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind
+Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O
+x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0
+x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void)
+u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1)
+(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def
+(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S
+O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead
+(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S
+O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O
+x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2
+(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl
+x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u)
+(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5
+t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3
+(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b))
+(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4
+H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10))
+H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead
+(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let
+H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0
+t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6
+(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4)))))))))))))))))
+(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w
+u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
+w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda
+(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2:
+C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead
+(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6
+\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_:
+(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g
+c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl
+t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda
+(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T
+t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))
+(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl)
+w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u
+t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v
+(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) |
+(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1
+t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let
+H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5)
+\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _)
+\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
+\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K
+(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead
+k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind
+T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2)
+\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v
+(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to
+((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u
+t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T
+(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to
+(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda
+(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3
+g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3
+g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda
+(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6:
+T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind
+Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
+(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda
+(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u
+x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0
+(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead
+(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2
+H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead
+(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1
+c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0)))))))))
+(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst)
+u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1
+(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6
+H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w
+v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow
+(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat
+Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _)
+\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead
+(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind
+Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to
+((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u
+t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T
+(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4)
+t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4)
+t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to
+((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u
+t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def
+(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall
+(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3
+(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5:
+T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12)
+in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g
+c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0)
+x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind
+Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda
+(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2
+(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u
+x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind
+Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2
+(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_:
+T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3
+g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
+(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u
+t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind
+Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1:
+T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
+(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0
+u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b)
+u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
+(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0
+H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u
+(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3
+H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead
+(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind
+Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0))))))))
+(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead
+(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0
+(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x
+H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0)
+(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0
+t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) |
+(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq
+T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w
+v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3)
+| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow
+t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v)
+H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
+| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1
+t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T
+(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to
+((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead
+(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3)
+v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to
+((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b
+Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5
+(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq
+B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda
+(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3:
+C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead
+(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def
+(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2
+(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead
+(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x:
+T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in
+(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u
+t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5:
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
+(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda
+(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5:
+T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0))))
+(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda
+(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind
+Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead
+(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2
+x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def
+(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3
+(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27)
+(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u
+t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u
+t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x)))
+(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O)
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S
+O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T
+T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind
+Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6:
+T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5:
+T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S
+O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u
+t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b)
+u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33:
+(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g
+(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O)
+t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
+(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
+(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0
+H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S
+O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat
+Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind
+Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b)
+u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17)
+(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2)
+u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind
+Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O)
+O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst
+(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead
+(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0))))
+(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O)
+O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0))
+(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead
+(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl
+v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O
+(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0))))
+(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w
+H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind
+Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u
+t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0)
+(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat
+Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0))
+(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b)
+u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31)))))))))
+(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind
+b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g
+Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind
+Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19
+(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7
+H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq
+T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T
+(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1
+t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v)
+H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to
+((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4
+H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3))
+(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind
+T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to
+((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0)
+\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl)
+w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast)
+u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
+F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
+(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g
+c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in
+(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2))))))))))))))))
+(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2
+t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0
+t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3
+t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0
+t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let
+H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda
+(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to
+(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow
+(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5
+t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to
+(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat
+Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g
+c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl
+t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat
+Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda
+(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T
+(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
+\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead
+(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead
+(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat
+Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to
+((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1
+t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat
+Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead
+(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda
+(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T
+(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda
+(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0
+t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T
+(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead
+(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv
+g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0
+(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat
+Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6
+t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2
+H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3)
+(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2)
+(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2
+t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1
+(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6
+H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda
+(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T
+(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2
+t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2
+H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1
+(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
+\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
+True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in
+(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0
+t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8
+H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T
+(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T
+(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1
+t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3
+t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2)
+\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast)
+t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow
+(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast)
+t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b)
+u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to
+((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7)))
+| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u
+t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7)
+\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in
+((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7
+_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7)
+in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5
+t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g
+c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4
+(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))
+(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3
+g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2
+t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0)
+(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4
+H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat
+Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0
+(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2
+H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead
+(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
+
+theorem ty3_sred_pr0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall
+(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g:
+G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1
+t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
+
+theorem ty3_sred_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall
+(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3:
+T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g:
+G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c:
+C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g:
+G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t
+(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
+
+theorem ty3_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g:
+G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g:
+G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3
+t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
+G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0
+(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t
+H2)))))))))))))) c t1 t2 H)))).
+
+theorem ty3_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall
+(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g:
+G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c
+t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3:
+(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2
+H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/pr3.ma".
+
+theorem ty3_cred_pr2:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind
+b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b:
+B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b)
+t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0
+(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0
+(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda
+(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0
+t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead
+c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl
+c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead
+c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0)
+(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1
+v2 H))))).
+
+theorem ty3_cred_pr3:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b:
+B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to
+(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b)
+t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1
+t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b:
+B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to
+(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0:
+T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b
+t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
+
+theorem ty3_gen_lift:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop
+h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2:
+T).(ty3 g e t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T
+(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e:
+C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x))
+(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c
+y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall
+(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x))
+(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall
+(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to
+(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t
+t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall
+(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda
+(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e:
+C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t))
+(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3:
+T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall
+(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e
+x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7:
+(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2:
+T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h
+x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda
+(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let
+H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda
+(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0
+t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda
+(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4)
+t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2
+H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift
+h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort
+m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort
+(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e
+(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
+T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1
+(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0
+(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind
+Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3:
+((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall
+(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2)
+t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda
+(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
+(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0
+t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
+(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
+(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n))
+x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n))
+t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))
+(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4)
+t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S
+n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift
+(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t)))
+(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift
+(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n)
+O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus
+x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4))
+(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g
+n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16)))))))))
+(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9)))
+H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n
+h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
+T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0
+(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
+T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift
+(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda
+(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h)))
+(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n
+(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0
+(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus
+x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n
+h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O
+x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n
+H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e
+d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2))
+x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst)
+u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall
+(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e:
+C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t))
+(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda
+(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
+(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0
+t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
+(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
+(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e
+(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h
+(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h
+(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2))))
+(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3
+d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2
+x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S
+n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift
+h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g
+e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus
+x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus
+(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3
+c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus
+n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2)))
+(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S
+n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O
+(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1
+(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst
+c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
+(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h)
+n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
+t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le
+(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T
+(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
+T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T
+(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O
+u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0
+O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
+O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h))
+(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h
+(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h))
+O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus
+O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h)))))
+(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0
+(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u
+t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b:
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b)
+u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift
+h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T
+(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4:
+T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5:
+(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6:
+(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
+(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1
+y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4:
+T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0
+(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9:
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda
+(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u
+t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda
+(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to
+(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda
+(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3
+g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2
+(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3)
+H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b)
+t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T
+u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1)
+x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0)
+e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4)
+t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let
+H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0
+x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda
+(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2)
+(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind
+b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16
+\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda
+(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3)))
+(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4:
+T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2
+x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e
+(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4:
+T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda
+(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0
+(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e
+(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0
+(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead
+e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b)
+x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
+(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))
+(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2)
+t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b)
+x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5))
+(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2
+c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind
+b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21))))
+(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u
+H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5)))))))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w
+u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
+c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0
+t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead
+(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0
+e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t))))
+(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1
+x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl)
+x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead
+(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2))))
+(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t)))
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11
+\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3
+(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall
+(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0:
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u))
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13
+\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in
+(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2:
+T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1
+(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
+(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16:
+(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6)
+in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u
+t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift
+h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t))))
+(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5:
+T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda
+(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3
+e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u
+(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
+(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst)
+x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall
+(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1)
+x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
+(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3
+e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2:
+T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
+(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
+t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda
+(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda
+(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25:
+(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7
+x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl)
+(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead
+(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))
+(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst)
+x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead
+(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7))
+(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h
+(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t)))
+(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind
+Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6))
+(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind
+Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1
+(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead
+(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4
+H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6)
+(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind
+Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9
+H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20)))))))))
+(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19)))))))
+(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0
+H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda
+(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to
+(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h
+x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0:
+T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall
+(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e
+x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T
+(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6:
+(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
+(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h
+x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda
+(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq
+T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1
+x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast)
+x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead
+(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind
+T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5
+x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3
+c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h
+x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3
+(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall
+(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0:
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t))
+(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13
+\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in
+(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0
+(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead
+(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g
+c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2
+(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4))
+\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0
+(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12
+(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3))
+e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2)))
+(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1
+t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead
+(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1
+x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1
+(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0
+(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4:
+T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda
+(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda
+(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1
+x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat
+Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2))
+(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1
+c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat
+Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e
+x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21)))))
+H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1
+H5))))))))))))))) c y x H0))))) H))))))).
+
+theorem ty3_tred:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T
+(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1:
+(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g
+c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))).
+
+theorem ty3_sconv_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
+u2) \to (pc3 c t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x:
+T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def
+(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g
+t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))).
+
+theorem ty3_sred_back:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c
+t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2
+t) \to (ty3 g c t1 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda
+(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
+(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t
+t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g
+c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t
+H1)))) (ty3_correct g c t2 t H1)))))))))).
+
+theorem ty3_sconv:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
+u2) \to (ty3 g c u1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda
+(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back
+g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/fwd.ma".
+
+include "LambdaDelta-1/pc3/fwd.ma".
+
+theorem ty3_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e
+t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c
+e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h
+d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3
+g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
+t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h
+d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5)
+(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m:
+nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop
+h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort
+(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0
+(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort
+(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
+(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h:
+nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0
+t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3:
+(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0
+(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le
+n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0)
+in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0
+(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind
+nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S
+n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0
+(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1
+(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0
+(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t)))
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
+d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T
+(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t))))
+(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef
+n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0
+(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda
+(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t))))
+(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x
+(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n))
+t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
+n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
+t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
+(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus
+h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n)
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
+(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda
+(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall
+(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u)
+(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h:
+nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0
+(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5
+\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3
+(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_:
+C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n)
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst)
+u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda
+(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop
+h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let
+H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S
+(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0
+h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0
+(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h
+(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S
+n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
+h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x
+d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S
+n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3
+g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift
+h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat
+d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S
+n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0
+(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus
+d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
+n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
+u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
+(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus
+h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n)
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
+(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
+t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3
+g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0
+(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
+(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0
+t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d
+u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b)
+(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u)
+(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead
+c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h
+d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead
+(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c:
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1:
+((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall
+(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0
+(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0:
+C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0
+c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v))
+(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind
+Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat
+Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat
+Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead
+(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat
+Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w)
+(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl
+g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t)
+(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift
+h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t))
+(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t))
+(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
+w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t)
+h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h
+d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda
+(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d
+t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3:
+((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
+(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3
+g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast)
+(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead
+(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0
+(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat
+Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4
+t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0))
+(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))).
+
+theorem ty3_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda
+(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda
+(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g
+c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g
+c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T
+(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m)))
+(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda
+(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3
+g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda
+(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n)
+(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind
+Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T
+(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n)
+(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda
+(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in
+(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T
+(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x:
+T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda
+(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g
+c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w:
+T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda
+(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0
+(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0:
+T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w
+(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u
+x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind
+Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead
+(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind
+Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0
+(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u
+t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t
+t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind
+Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0
+(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10:
+(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w
+(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1
+(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0
+H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g
+c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3:
+(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T
+(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat
+Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T
+(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x
+t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))).
+
+theorem ty3_unique:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0:
+T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall
+(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0
+t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s
+c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g
+c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t:
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0
+t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n)
+t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1)))))
+(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0
+(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g
+e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda
+(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1
+x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n
+c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n
+H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
+(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind
+Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr)
+x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0)
+\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
+T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def
+(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
+(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d
+H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d
+H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O
+(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9)))))))))
+H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst)
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))
+(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def
+(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead
+x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0
+(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst)
+x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1)
+H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4))
+(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind
+Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_:
+((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2:
+T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4:
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O
+u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3
+c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr)
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind
+Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in
+C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0
+(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2))))
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1
+t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O
+x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7:
+(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal
+C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0)
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
+x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match
+e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
+t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
+T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def
+(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
+(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in
+(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind
+Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3
+g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2
+n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda
+(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to
+(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3:
+T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2
+t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0)
+t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda
+(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead
+(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0
+(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3
+g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead
+(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5))))))
+(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w:
+T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2:
+T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3:
+((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t)
+t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v)
+t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0:
+T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_:
+T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t))
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead
+(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl)
+w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0
+t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3
+(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2
+H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to
+(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda
+(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4:
+T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda
+(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0
+t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4)
+(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2)
+t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t
+(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0
+(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4
+H4)))))))))))) c u t1 H))))).
+
+theorem ty3_gen_abst_abst:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
+(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2
+T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst)
+u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u
+t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T
+(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u)
+t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2)
+x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst)
+u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda
+(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
+x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind
+Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c
+(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda
+(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind
+Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3
+g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5:
+(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y
+\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g
+c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5
+(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y))))))))
+(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g
+Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind
+Abst) u t2) H))))))).
+
+theorem ty3_typecheck:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t
+v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H:
+(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u:
+T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g
+c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))
+(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v
+H)))))).
+
+theorem ty3_getl_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t
+t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2)
+\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda
+(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0
+t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
+nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3:
+T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b:
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b)
+v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0:
+(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v:
+T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m
+H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
+T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
+w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n
+i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w)))
+(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7
+\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v)))
+H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1:
+C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
+(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
+(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2)
+\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono
+c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12:
+(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v
+(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T
+u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def
+(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d
+H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w))))
+(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0)
+u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0
+H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n
+H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall
+(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall
+(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to
+(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda
+(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda
+(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0
+(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T
+(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq
+T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0:
+nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C
+(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind
+b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7))
+in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B
+(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])]))
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0
+(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d
+d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind
+b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w:
+T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n
+c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T
+(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0:
+B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda
+(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9))))))
+(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1:
+((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
+nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v:
+T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda
+(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda
+(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0:
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind
+b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
+(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3
+(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w:
+T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead
+(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
+u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T
+t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0
+d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind
+b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T
+(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
+(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
+T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s
+(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0
+(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
+x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
+v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0
+u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
+(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0:
+T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b:
+B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex
+T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3:
+((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to
+(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind
+b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v
+t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda
+(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0)))
+(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8:
+(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead
+(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex
+T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v
+x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0:
+T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1
+(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9:
+(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1
+H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4)))))))))))))))))))
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1
+t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i:
+nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v:
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
+w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3:
+((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0:
+T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
+Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda
+(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T
+t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T
+(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4:
+T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat
+Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2:
+T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat
+Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w:
+T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x
+t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6))
+(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4)))
+(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda
+(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s
+(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
+T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s
+(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6))
+(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T
+(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2
+x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v
+H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4))))))))))))))))))
+c t u H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/pr3_props.ma".
+
+include "LambdaDelta-1/sty0/fwd.ma".
+
+theorem ty3_sty0:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_:
+T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda
+(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3:
+((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_:
+(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0
+H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda
+(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in
+(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in
+(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t))
+(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
+u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall
+(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda
+(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in
+(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C
+T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g
+e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift
+(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O
+t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8:
+(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
+(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal
+C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
+(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
+x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match
+e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
+t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0:
+T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def
+(eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 u0 H12) in (let H16
+\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0)))
+H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(sty0 g c1 u0
+x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2
+H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O
+u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq
+T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
+(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind
+C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match
+b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
+\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
+False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0)
+n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift
+(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0
+t)).(\lambda (_: ((\forall (t2: T).((sty0 g d u0 t2) \to (ty3 g d u0
+t2))))).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 (TLRef n) t2)).(let H_x
+\def (sty0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1
+t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
+O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2)
+(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1
+t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
+O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_:
+(sty0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift
+(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C
+(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind
+Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr)
+x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
+with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
+(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0
+(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind
+(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda
+(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0:
+T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T
+t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0
+(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda
+(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1
+x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T
+(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1)
+(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d
+(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
+(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead
+d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0)
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
+x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def
+(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0
+H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7
+u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O
+t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1
+(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1:
+C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1
+H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall
+(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2
+t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4)
+\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4:
+(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0
+t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead
+c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4)))
+(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g
+(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0
+x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x)
+H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead
+(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8)))))
+H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda
+(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to
+(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
+(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2)
+\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead
+(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5
+\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3:
+T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2)
+(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead
+(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead
+(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0:
+T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9
+\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T
+(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T
+(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T
+(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead
+(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12:
+(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_:
+T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3
+g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead
+(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0
+(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda
+(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat
+Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w
+u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind
+Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl)
+w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v
+t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead
+(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2
+H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0
+w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5))))))))))))))
+(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2
+t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2
+t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3:
+((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4:
+T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def
+(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T
+(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda
+(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0
+t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def
+(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r
+T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3
+t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10
+\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0
+x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1))
+(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t:
+T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0
+x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead
+(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g
+c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat
+Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1
+(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y))))
+(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/props.ma".
+
+include "LambdaDelta-1/pc3/subst1.ma".
+
+include "LambdaDelta-1/getl/getl.ma".
+
+theorem ty3_gen_cabbr:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
+(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
+e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e:
+C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0))
+\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0:
+T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr)
+u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a:
+C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7)
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda
+(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d
+u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e
+u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d
+x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a
+x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9
+H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0
+H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0:
+C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
+d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort
+m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort
+(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t:
+T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort
+m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
+T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m)))
+(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g
+a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t:
+T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0:
+T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0:
+C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e:
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0:
+nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)))
+(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0
+(le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6))
+in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr)
+u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1
+(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let
+H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d
+(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11
+\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in
+(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind
+Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u
+u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2)))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind
+Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14:
+(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1:
+C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind
+nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S
+n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18:
+(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S
+n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S
+n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2
+e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u
+(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift
+(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S
+n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S
+n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0
+(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4
+x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2
+(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r
+nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S
+n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O)
+n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
+u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5)
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0))
+(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O)
+d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda
+(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0))
+(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n)
+H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n)))
+(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus
+d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0
+H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt
+Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11))))))
+(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda
+(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S
+O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0:
+nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0
+(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind
+nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
+n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d
+(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0)
+(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
+(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind
+Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u)
+(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e
+(Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T
+u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let
+H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in
+(eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda
+(c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift
+n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T
+(lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0))
+(subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n
+(plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0:
+T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift
+(S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n)))
+(ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge
+n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12)))))
+H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n
+(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S
+O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O))
+(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift
+(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O
+t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O))
+(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O)))
+t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0
+(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus
+d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0
+u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0
+(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0)
+(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0:
+nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S
+O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0
+n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a
+(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
+(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
+n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
+O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
+O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
+H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
+C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0))
+\to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O)
+d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda
+(H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4:
+(csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0
+a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0
+u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
+d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat
+(minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e
+(Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d
+(Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n)))
+(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0
+(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
+(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u)
+x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n)
+(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0
+(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst)
+d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda
+(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_:
+C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1
+(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d
+x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1
+(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop
+(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in
+(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0
+(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0)))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u)
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus
+d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda
+(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0
+(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0
+(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind
+Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u
+(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
+(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
+x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n))
+x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda
+(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S
+n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0:
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S
+n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
+u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O)
+(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
+a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0:
+T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0
+(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S
+O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S
+n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n))
+x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O)
+(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n)
+(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5
+H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0
+H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus
+d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead
+d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r
+nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def
+(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let
+H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr)
+u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C
+(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind
+Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0)
+H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
+with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
+(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S
+O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u)
+(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n
+(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S
+O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O))
+(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift
+(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O
+u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O))
+(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O)))
+t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0
+(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus
+d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0
+u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0
+(lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0)
+(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0:
+nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S
+O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0
+n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a
+(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
+(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
+n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
+O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
+O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
+H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
+C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b:
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d:
+nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall
+(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop
+(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
+(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5:
+(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let
+H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead
+(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
+d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8:
+(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d
+x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head
+(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d
+x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b)
+x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
+t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S
+O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d)
+x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
+t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b)
+(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead
+(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3
+(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b
+x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S
+d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head
+u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S
+O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1
+H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0:
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1:
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
+(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a:
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
+C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
+(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5:
+(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let
+H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d
+x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d
+x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6)
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w
+(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d
+x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind
+Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S
+O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u
+x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S
+O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y:
+T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst)
+x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5
+(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1
+(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4
+(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25
+\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6
+x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead
+(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in
+(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat
+Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead
+(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl)
+(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead
+(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v
+(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat
+Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift
+(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d
+x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind
+Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S
+d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t)
+t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t
+(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7))
+(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead
+(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d))
+(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S
+O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda
+(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e
+(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4
+t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl
+d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
+nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
+C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S
+O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda
+(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda
+(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda
+(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let
+H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead
+(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d
+u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def
+(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift
+t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast)
+x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d
+x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3)
+t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2)
+H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d))
+(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda
+(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S
+O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat
+Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1
+H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
+
+theorem ty3_gen_cvoid:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
+(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
+e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e:
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to
+(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u
+t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl
+d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4
+t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d
+c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0
+a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9:
+(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def
+(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in
+(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d
+x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S
+O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0)
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0
+d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15:
+(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d
+x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0:
+T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3
+(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15)
+in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0))
+H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift
+(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1
+H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u
+H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda
+(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e
+(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0
+a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m))
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T
+(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m
+(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g
+m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m)))
+(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl
+d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0:
+T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void)
+u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt
+n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0
+(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e
+(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n)
+d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind
+nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S
+n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
+(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
+(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
+\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
+(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
+(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
+(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
+(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
+T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def
+(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0)
+u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n))
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n))
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0)
+(lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus
+d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t
+(\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S
+O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3)
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
+t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16
+x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O)
+(plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0:
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O)
+d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O
+x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0
+(lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0
+(TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift
+(S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3))
+(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t
+H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0
+(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0
+(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r
+nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in
+(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
+T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0
+(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d
+(Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda
+(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0:
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O)
+d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O)
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat
+(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S
+O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq
+T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
+(lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
+(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t)
+(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T
+(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n
+(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus
+n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O
+t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O
+t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O
+n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S
+O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t)))
+(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr)
+u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le
+n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n
+(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym
+(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus
+O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
+H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
+C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0))
+\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0:
+nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a:
+C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6
+\def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind
+Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0)
+c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S
+(minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0
+(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n))))
+(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
+(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
+(Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
+\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
+(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
+(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
+(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
+(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
+T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T
+(lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus
+d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S
+n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda
+(_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0))
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0
+(S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S
+O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def
+(eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0))
+H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2
+(\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S
+n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n)
+O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
+(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
+T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S
+n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0
+H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0
+H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift
+(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S
+n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8))))))))
+(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda
+(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S
+O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0:
+nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n
+(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
+n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
+u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl
+n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n
+H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind
+Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind
+Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0)
+H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
+n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
+u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S
+O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift
+(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus
+(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S
+O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda
+(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef
+(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O))))
+(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T
+(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0))
+(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n
+(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0)))
+(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n
+(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge
+n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0)
+(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1)
+n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n
+(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O))))
+(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n
+(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda
+(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
+(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3:
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind
+b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0
+(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e:
+C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind
+Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def
+(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
+u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S
+O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
+(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda
+(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
+(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in
+(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1)))
+H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0:
+T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0
+(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0
+(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1
+y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda
+(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in
+(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda
+(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0
+(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0))
+(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T
+T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda
+(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S
+O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S
+O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17:
+(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3)
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead
+(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O)
+(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
+x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind
+b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
+x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead
+(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2))
+(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3))
+(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3
+(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u
+H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
+(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall
+(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
+(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a:
+C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind
+Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
+(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5:
+(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T
+(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
+Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O)
+d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d
+x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0:
+T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in
+(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
+Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead
+(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u
+(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14
+\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2
+x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
+Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda
+(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0
+(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in
+(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
+(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18:
+(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4
+x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r
+T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18))
+in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d
+x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d
+(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind
+Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
+a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst)
+x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4
+(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4
+(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4
+x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2
+x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d
+x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind
+Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d
+x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2)
+(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u
+H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7)))))))
+H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall
+(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3:
+((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind
+Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda
+(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a:
+C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8:
+(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
+(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in
+(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def
+(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O)
+d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0:
+C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void)
+u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4
+(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T
+(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O)
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def
+(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
+t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d
+x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S
+O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
+(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda
+(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2
+x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d
+x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda
+(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast)
+(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O)
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def
+(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d
+H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1)
+(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0))
+(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O)
+d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
+a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S
+O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2)
+(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0
+x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a
+x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0))
+(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S
+O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0
+H8))))))) H6)))))))))))))))) c t1 t2 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr0/defs.ma".
+
+include "LambdaDelta-1/C/defs.ma".
+
+inductive wcpr0: C \to (C \to Prop) \def
+| wcpr0_refl: \forall (c: C).(wcpr0 c c)
+| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall
+(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k
+u1) (CHead c2 k u2)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/wcpr0/defs.ma".
+
+theorem wcpr0_gen_sort:
+ \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort
+n))))
+\def
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n)
+x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C
+x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c:
+C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c:
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0))
+(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda
+(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
+c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C
+(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
+(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1))
+H5))))))))))) y x H0))) H))).
+
+theorem wcpr0_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
+(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
+(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c:
+C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2:
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda
+(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0:
+C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2:
+C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))
+(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal
+C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1)
+(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
+T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1
+c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C
+(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
+T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))
+(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to
+(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead
+c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2:
+T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0
+k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
+\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
+(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K
+k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C
+(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3:
+T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
+T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10
+\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1
+(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda
+(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
+(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k
+u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C
+c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def
+(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1
+(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda
+(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
+(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T
+(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3))))
+(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq
+C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0
+c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C
+(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x
+H0))) H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/wcpr0/defs.ma".
+
+include "LambdaDelta-1/getl/props.ma".
+
+theorem wcpr0_drop:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
+c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1)
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _)
+\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k
+u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0)
+(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1)
+(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k
+u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t:
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2)
+(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7))))
+H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0
+u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1:
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead
+e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
+(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b:
+B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
+(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1
+k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
+O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0
+x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
+c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop
+(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f:
+F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
+(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead
+e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead
+e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
+(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0
+u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
+(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1
+(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k)
+h)))))))))) c1 c2 H))).
+
+theorem wcpr0_drop_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0
+u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
+c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2)
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _)
+\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k
+u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0)
+(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2)
+(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k
+u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t:
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1)
+(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7))))
+H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0
+u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1:
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead
+e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
+(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b:
+B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
+(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1
+k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
+O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1
+u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
+c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop
+(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f:
+F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
+(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead
+e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead
+e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
+(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0
+x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
+(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1
+(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k)
+h)))))))))) c2 c1 H))).
+
+theorem wcpr0_getl:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k
+u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1)
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1
+u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda
+(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3
+(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
+(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let
+H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
+u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in
+(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K
+(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
+O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r
+T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
+(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3)))))
+(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1
+u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4
+(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2
+(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
+F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5
+\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3)
+(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda
+(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
+u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f)
+u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
+x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
+(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
+u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k
+(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
+K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
+K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
+u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
+(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
+u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1)
+(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
+(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
+u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
+u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
+(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7:
+(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2)
+H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat
+f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
+T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
+u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
+e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
+(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda
+(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2)
+H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
+
+theorem wcpr0_getl_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
+u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k
+u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2)
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1
+u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda
+(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4
+(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
+(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let
+H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
+u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in
+(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K
+(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
+O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r
+T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
+(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t)))))
+(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
+u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3
+(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1
+(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
+F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5
+\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4)
+(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda
+(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
+u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f)
+u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
+e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
+(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
+u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k
+(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
+K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
+K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
+u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
+(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
+u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2)
+(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
+(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
+u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
+u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
+(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7:
+(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1)
+H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat
+f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
+T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
+u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
+e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
+(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda
+(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1)
+H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/wf3/fwd.ma".
+
+theorem wf3_clear_conf:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall
+(c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
+\def
+ \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind
+(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0
+c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u:
+T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u)
+c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e
+c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g
+c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def
+(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
+
+theorem clear_wf3_trans:
+ \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall
+(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda
+(c2: C).(clear c2 d2))))))))
+\def
+ \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0
+d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2
+d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g:
+G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x
+\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T
+(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda
+(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g
+e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
+\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2))
+(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2:
+C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2:
+C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u
+w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2
+(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e
+(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e
+x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
+(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2:
+C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u)
+c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u)
+(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda
+(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
+\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void)
+(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w:
+T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind
+b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3:
+(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e
+x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C
+(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3
+g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C
+(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2
+(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void
+g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1)))))))))
+(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
+((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f:
+F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c
+d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda
+(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2:
+C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda
+(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C
+(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2
+d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/ty3/defs.ma".
+
+inductive wf3 (g: G): C \to (C \to Prop) \def
+| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
+| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind
+b) u) (CHead c2 (Bind b) u))))))))
+| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
+| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
+T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/wf3/defs.ma".
+
+theorem wf3_gen_sort1:
+ \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to
+(eq C x (CSort m)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort
+m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c:
+C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda
+(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0:
+nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat
+(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n)
+\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in
+(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C
+(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4:
+(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1
+(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m)
+H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
+T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
+B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind
+C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O))
+(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C
+c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat
+f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee:
+C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C
+c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
+
+theorem wf3_gen_bind1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b:
+B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2:
+C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda
+(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3
+C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b:
+B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind
+b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda
+(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2:
+C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O))))
+(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or
+(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v))))
+(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w:
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void)
+(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w:
+T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C
+(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v)
+H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C
+(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1
+c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2:
+C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g
+c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2))))
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
+(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0
+u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1
+(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
+c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
+b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind
+b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9:
+(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v))))
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w:
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u)
+(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda
+(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind
+T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0:
+T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b)
+t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
+C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1
+H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b)
+v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v)
+(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
+C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
+False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2
+(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
+c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C
+(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5)))))))))))
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
+(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
+w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g
+c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind
+b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
+(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v)
+H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
+(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4)
+in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0
+b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t:
+T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def
+(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10
+c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind
+b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void)
+(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
+c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda
+(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind
+Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g
+c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2
+(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6))
+H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0
+c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T
+(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
+g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C
+(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead
+c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C
+T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
+g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
+c1 v w) \to False))))) H4))))))))) y x H0))) H)))))).
+
+theorem wf3_gen_flat1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f:
+F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f:
+F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat
+f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y:
+C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0:
+C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m:
+nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def
+(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m))
+H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda
+(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4:
+(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C
+(CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1
+(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v))
+\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0
+u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u)
+(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2
+(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f)
+v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C
+(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead
+c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in
+C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _)
+\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead
+c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda
+(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0
+(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8)
+in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in
+H10))))) H5)) H4))))))))) y x H0))) H)))))).
+
+theorem wf3_gen_head2:
+ \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k:
+K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k:
+K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda
+(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind
+b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_:
+C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
+k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k
+v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b:
+B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b:
+B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3
+g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c
+k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2
+(Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in
+(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def
+(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda
+(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2
+(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k
+(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0
+(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B
+(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K
+(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6))
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
+k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u
+t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void)
+(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in
+((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow
+k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2
+(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void)
+k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0:
+C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0))))))
+H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c
+H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v))
+\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in
+(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind
+b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind
+Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12
+(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))
+Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_:
+T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def
+(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind
+C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq
+K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda
+(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k
+v))))))))))))) x y H0))) H)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/wf3/clear.ma".
+
+include "LambdaDelta-1/ty3/dec.ma".
+
+theorem wf3_getl_conf:
+ \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall
+(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2:
+C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2)))))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1:
+C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to
+(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g
+d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v)))
+(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1:
+C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda
+(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda
+(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v)
+(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def
+(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T
+(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
+(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3
+g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
+g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
+b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda
+(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1
+v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_:
+C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda
+(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind
+b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O
+(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))
+x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3:
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1
+c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to
+False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
+g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2
+(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6:
+((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind
+Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in
+(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0
+(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H:
+((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
+b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind
+(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0:
+nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0)
+(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g
+(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort
+n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda
+(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead
+d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to
+(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl
+(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3
+g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r
+k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0:
+B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r
+(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t
+b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda
+(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_:
+T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
+C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to
+False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v)))
+(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3:
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3:
+C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t
+w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
+(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_:
+C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda
+(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind
+b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g
+x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2:
+C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
+n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11)))
+c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3
+(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_:
+C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3:
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3))
+(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda
+(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
+d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort
+O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t
+w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0:
+C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda
+(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let
+H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b)
+v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n)
+(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
+n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
+C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12
+(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda
+(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c
+(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1
+v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n
+H1)))))))))))))) c1)))) i)).
+
+theorem getl_wf3_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2
+d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1
+d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x
+\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x
+in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2))
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2)))
+(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x
+d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2
+d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n:
+nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1:
+C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall
+(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda
+(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda
+(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C
+(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2
+d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c
+d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda
+(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k
+t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind
+(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g
+(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b:
+B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2
+H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda
+(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t)
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3
+g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in
+(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall
+(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c
+(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T
+(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2))
+(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2:
+C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t
+x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda
+(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9
+b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2:
+T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead
+c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void)
+(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort
+O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c
+d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C
+(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C
+(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S
+n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S
+n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2))
+(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4)))))
+k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/wf3/ty3.ma".
+
+include "LambdaDelta-1/app/defs.ma".
+
+theorem wf3_mono:
+ \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall
+(c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c
+c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0
+c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0:
+(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C
+(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2
+H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2
+c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3
+c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u
+t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b)
+u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in
+(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind
+b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u)
+c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead
+c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda
+(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4:
+C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4:
+C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2
+u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2
+x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
+(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind
+b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0
+H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind
+Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall
+(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0
+(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda
+(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b)
+u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort
+O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u
+w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4:
+C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def
+H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort
+O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3:
+C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4)
+\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g
+c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g
+(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in
+(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C
+c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4)))
+(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq
+C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4))
+(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3
+(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda
+(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_:
+T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u
+w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4
+(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
+C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O))
+c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind
+b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r
+C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort
+O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C
+(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6))))))
+H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void)
+(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w:
+T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
+C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void)
+(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void)
+(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3
+g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda
+(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead
+c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K
+(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4))))))))))))
+(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1:
+((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u:
+T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f)
+u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y))))))))))
+c c1 H)))).
+
+theorem wf3_total:
+ \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
+C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
+(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
+(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
+H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
+(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
+(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
+(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
+T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
+False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
+g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
+(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
+b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
+False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
+(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
+(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
+(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
+
+theorem ty3_shift1:
+ \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall
+(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c
+t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c
+(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall
+(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0
+t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0:
+C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3
+g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2))))))))
+(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C
+c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g
+(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda
+(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C
+(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u)
+(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0:
+C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to
+(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8
+\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T
+(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1))
+(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x:
+T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1)
+(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2
+H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5)))))))))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2:
+(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to
+(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u:
+T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
+B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort
+O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind
+b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
+in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
+(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
+in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9:
+(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b
+(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in
+(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u)))
+(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12
+\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11
+(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0:
+T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O)
+(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1
+(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def
+(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall
+(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1
+t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g
+c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void)
+(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort
+O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda
+(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1)
+(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g
+c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12))))
+(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b
+H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall
+(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
+t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1
+(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead
+c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1
+(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1
+c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort
+(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in
+(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1
+(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in
+(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3
+g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
+t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def
+(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I
+(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u
+t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0)))
+H))).
+
+theorem wf3_idem:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
+c2 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
+c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
+nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
+c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
+(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
+c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
+(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
+((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
+c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
+c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
+
+theorem wf3_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
+u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
+C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
+(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
+x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
+u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/wf3/getl.ma".
+
+theorem wf3_pr2_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
+u) \to (pr2 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2
+c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u:
+T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
+C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3
+u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def
+(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in
+(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x:
+T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0
+g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t)
+(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda
+(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7)))))
+H5)))))))))))))))))) c1 t1 t2 H))))).
+
+theorem wf3_pr3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
+u) \to (pr3 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall
+(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t
+t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1
+c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
+(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1
+t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2
+H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
+
+theorem wf3_pc3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1
+u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda
+(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g
+c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
+(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4:
+(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf
+g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2)))))
+H3)))))))))))).
+
+theorem wf3_ty3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall
+(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4
+t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0:
+T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4
+x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4
+c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c:
+C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g
+c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
+(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g
+c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
+(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u
+t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
+(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1:
+((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b:
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u)
+t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to
+(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g
+c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t
+H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c
+c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
+c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/preamble.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
+include "LambdaDelta-2/T/props.ma".
+
+inline procedural "LambdaDelta-1/C/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/preamble.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
+inline procedural "LambdaDelta-1/T/dec.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/preamble.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
+inline procedural "LambdaDelta-1/T/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/asucc/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/aplus/defs.ma".
+
+include "LambdaDelta-2/next_plus/props.ma".
+
+inline procedural "LambdaDelta-1/aplus/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/A/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/aprem/defs.ma".
+
+inline procedural "LambdaDelta-1/aprem/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/aprem/fwd.ma".
+
+include "LambdaDelta-2/leq/defs.ma".
+
+inline procedural "LambdaDelta-1/aprem/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/arity/props.ma".
+
+include "LambdaDelta-2/arity/cimp.ma".
+
+include "LambdaDelta-2/aprem/props.ma".
+
+inline procedural "LambdaDelta-1/arity/aprem.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/arity/defs.ma".
+
+include "LambdaDelta-2/cimp/props.ma".
+
+inline procedural "LambdaDelta-1/arity/cimp.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/leq/defs.ma".
+
+include "LambdaDelta-2/getl/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/arity/defs.ma".
+
+include "LambdaDelta-2/leq/asucc.ma".
+
+include "LambdaDelta-2/getl/drop.ma".
+
+inline procedural "LambdaDelta-1/arity/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/arity/props.ma".
+
+include "LambdaDelta-2/drop1/fwd.ma".
+
+inline procedural "LambdaDelta-1/arity/lift1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csuba/arity.ma".
+
+include "LambdaDelta-2/pr3/defs.ma".
+
+include "LambdaDelta-2/pr1/defs.ma".
+
+include "LambdaDelta-2/wcpr0/getl.ma".
+
+include "LambdaDelta-2/pr0/fwd.ma".
+
+include "LambdaDelta-2/arity/subst0.ma".
+
+inline procedural "LambdaDelta-1/arity/pr3.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/arity/fwd.ma".
+
+inline procedural "LambdaDelta-1/arity/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/arity/props.ma".
+
+include "LambdaDelta-2/fsubst0/fwd.ma".
+
+include "LambdaDelta-2/csubst0/getl.ma".
+
+include "LambdaDelta-2/subst0/dec.ma".
+
+include "LambdaDelta-2/subst0/fwd.ma".
+
+include "LambdaDelta-2/getl/getl.ma".
+
+inline procedural "LambdaDelta-1/arity/subst0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/A/defs.ma".
+
+include "LambdaDelta-2/G/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/asucc/defs.ma".
+
+inline procedural "LambdaDelta-1/asucc/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/cimp/defs.ma".
+
+include "LambdaDelta-2/getl/getl.ma".
+
+inline procedural "LambdaDelta-1/cimp/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/clear/fwd.ma".
+
+include "LambdaDelta-2/drop/fwd.ma".
+
+inline procedural "LambdaDelta-1/clear/drop.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/clear/defs.ma".
+
+inline procedural "LambdaDelta-1/clear/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/clear/fwd.ma".
+
+inline procedural "LambdaDelta-1/clear/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
+include "LambdaDelta-2/s/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/clen/defs.ma".
+
+include "LambdaDelta-2/getl/props.ma".
+
+inline procedural "LambdaDelta-1/clen/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/cnt/defs.ma".
+
+include "LambdaDelta-2/lift/fwd.ma".
+
+inline procedural "LambdaDelta-1/cnt/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csuba/getl.ma".
+
+include "LambdaDelta-2/csuba/props.ma".
+
+include "LambdaDelta-2/arity/props.ma".
+
+include "LambdaDelta-2/csubv/getl.ma".
+
+inline procedural "LambdaDelta-1/csuba/arity.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csuba/defs.ma".
+
+include "LambdaDelta-2/clear/fwd.ma".
+
+inline procedural "LambdaDelta-1/csuba/clear.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/arity/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csuba/fwd.ma".
+
+include "LambdaDelta-2/drop/fwd.ma".
+
+inline procedural "LambdaDelta-1/csuba/drop.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csuba/defs.ma".
+
+inline procedural "LambdaDelta-1/csuba/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csuba/drop.ma".
+
+include "LambdaDelta-2/csuba/clear.ma".
+
+include "LambdaDelta-2/getl/clear.ma".
+
+inline procedural "LambdaDelta-1/csuba/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csuba/defs.ma".
+
+inline procedural "LambdaDelta-1/csuba/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/csuba.ma".
+
+inline procedural "LambdaDelta-1/csubc/arity.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubc/clear.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/defs.ma".
+
+include "LambdaDelta-2/sc3/props.ma".
+
+inline procedural "LambdaDelta-1/csubc/csuba.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sc3/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/fwd.ma".
+
+include "LambdaDelta-2/sc3/props.ma".
+
+inline procedural "LambdaDelta-1/csubc/drop.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/drop.ma".
+
+inline procedural "LambdaDelta-1/csubc/drop1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/defs.ma".
+
+inline procedural "LambdaDelta-1/csubc/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/drop.ma".
+
+include "LambdaDelta-2/csubc/clear.ma".
+
+inline procedural "LambdaDelta-1/csubc/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/defs.ma".
+
+include "LambdaDelta-2/sc3/props.ma".
+
+inline procedural "LambdaDelta-1/csubc/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst0/props.ma".
+
+include "LambdaDelta-2/csubst0/fwd.ma".
+
+include "LambdaDelta-2/clear/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubst0/clear.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/defs.ma".
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst0/fwd.ma".
+
+include "LambdaDelta-2/drop/fwd.ma".
+
+include "LambdaDelta-2/s/props.ma".
+
+inline procedural "LambdaDelta-1/csubst0/drop.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst0/defs.ma".
+
+inline procedural "LambdaDelta-1/csubst0/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst0/clear.ma".
+
+include "LambdaDelta-2/csubst0/drop.ma".
+
+include "LambdaDelta-2/getl/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubst0/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst0/defs.ma".
+
+inline procedural "LambdaDelta-1/csubst0/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst0/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst1/defs.ma".
+
+include "LambdaDelta-2/csubst0/fwd.ma".
+
+include "LambdaDelta-2/subst1/props.ma".
+
+inline procedural "LambdaDelta-1/csubst1/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst1/props.ma".
+
+include "LambdaDelta-2/csubst0/getl.ma".
+
+include "LambdaDelta-2/subst1/props.ma".
+
+include "LambdaDelta-2/drop/props.ma".
+
+inline procedural "LambdaDelta-1/csubst1/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst1/defs.ma".
+
+include "LambdaDelta-2/subst1/defs.ma".
+
+inline procedural "LambdaDelta-1/csubst1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/defs.ma".
+
+include "LambdaDelta-2/clear/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubt/clear.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/arity.ma".
+
+inline procedural "LambdaDelta-1/csubt/csuba.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/fwd.ma".
+
+include "LambdaDelta-2/drop/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubt/drop.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/defs.ma".
+
+inline procedural "LambdaDelta-1/csubt/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/clear.ma".
+
+include "LambdaDelta-2/csubt/drop.ma".
+
+include "LambdaDelta-2/getl/clear.ma".
+
+inline procedural "LambdaDelta-1/csubt/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/getl.ma".
+
+include "LambdaDelta-2/pc3/left.ma".
+
+inline procedural "LambdaDelta-1/csubt/pc3.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/defs.ma".
+
+inline procedural "LambdaDelta-1/csubt/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/pc3.ma".
+
+include "LambdaDelta-2/csubt/props.ma".
+
+inline procedural "LambdaDelta-1/csubt/ty3.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubv/defs.ma".
+
+include "LambdaDelta-2/clear/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubv/clear.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubv/props.ma".
+
+include "LambdaDelta-2/drop/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubv/drop.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubv/clear.ma".
+
+include "LambdaDelta-2/csubv/drop.ma".
+
+include "LambdaDelta-2/getl/fwd.ma".
+
+inline procedural "LambdaDelta-1/csubv/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubv/defs.ma".
+
+include "LambdaDelta-2/T/props.ma".
+
+inline procedural "LambdaDelta-1/csubv/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
+include "LambdaDelta-2/lift/defs.ma".
+
+include "LambdaDelta-2/r/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/drop/defs.ma".
+
+inline procedural "LambdaDelta-1/drop/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/drop/fwd.ma".
+
+include "LambdaDelta-2/lift/props.ma".
+
+include "LambdaDelta-2/r/props.ma".
+
+inline procedural "LambdaDelta-1/drop/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/drop/defs.ma".
+
+include "LambdaDelta-2/lift1/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/drop1/defs.ma".
+
+inline procedural "LambdaDelta-1/drop1/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/drop1/fwd.ma".
+
+include "LambdaDelta-2/getl/drop.ma".
+
+inline procedural "LambdaDelta-1/drop1/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/drop1/fwd.ma".
+
+include "LambdaDelta-2/drop/props.ma".
+
+include "LambdaDelta-2/getl/defs.ma".
+
+inline procedural "LambdaDelta-1/drop1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/A/defs.ma".
+
+include "LambdaDelta-2/G/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ex0/defs.ma".
+
+include "LambdaDelta-2/leq/defs.ma".
+
+include "LambdaDelta-2/aplus/props.ma".
+
+inline procedural "LambdaDelta-1/ex0/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ex1/defs.ma".
+
+include "LambdaDelta-2/ty3/fwd.ma".
+
+include "LambdaDelta-2/pc3/fwd.ma".
+
+include "LambdaDelta-2/nf2/pr3.ma".
+
+include "LambdaDelta-2/nf2/props.ma".
+
+include "LambdaDelta-2/arity/defs.ma".
+
+include "LambdaDelta-2/leq/props.ma".
+
+inline procedural "LambdaDelta-1/ex1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ex2/defs.ma".
+
+include "LambdaDelta-2/nf2/defs.ma".
+
+include "LambdaDelta-2/pr2/fwd.ma".
+
+include "LambdaDelta-2/arity/fwd.ma".
+
+inline procedural "LambdaDelta-1/ex2/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/flt/defs.ma".
+
+include "LambdaDelta-2/C/props.ma".
+
+inline procedural "LambdaDelta-1/flt/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubst0/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/fsubst0/defs.ma".
+
+inline procedural "LambdaDelta-1/fsubst0/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/props.ma".
+
+include "LambdaDelta-2/clear/drop.ma".
+
+inline procedural "LambdaDelta-1/getl/clear.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/props.ma".
+
+inline procedural "LambdaDelta-1/getl/dec.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/drop/defs.ma".
+
+include "LambdaDelta-2/clear/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/props.ma".
+
+include "LambdaDelta-2/clear/drop.ma".
+
+inline procedural "LambdaDelta-1/getl/drop.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/fwd.ma".
+
+include "LambdaDelta-2/clear/props.ma".
+
+include "LambdaDelta-2/flt/props.ma".
+
+inline procedural "LambdaDelta-1/getl/flt.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/defs.ma".
+
+include "LambdaDelta-2/drop/fwd.ma".
+
+include "LambdaDelta-2/clear/fwd.ma".
+
+inline procedural "LambdaDelta-1/getl/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/drop.ma".
+
+include "LambdaDelta-2/getl/clear.ma".
+
+inline procedural "LambdaDelta-1/getl/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/getl/fwd.ma".
+
+include "LambdaDelta-2/drop/props.ma".
+
+include "LambdaDelta-2/clear/props.ma".
+
+inline procedural "LambdaDelta-1/getl/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/iso/defs.ma".
+
+include "LambdaDelta-2/tlist/defs.ma".
+
+inline procedural "LambdaDelta-1/iso/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/iso/fwd.ma".
+
+inline procedural "LambdaDelta-1/iso/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/leq/props.ma".
+
+inline procedural "LambdaDelta-1/leq/asucc.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/aplus/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/leq/defs.ma".
+
+inline procedural "LambdaDelta-1/leq/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/leq/fwd.ma".
+
+include "LambdaDelta-2/aplus/props.ma".
+
+inline procedural "LambdaDelta-1/leq/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/tlist/defs.ma".
+
+include "LambdaDelta-2/s/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift/defs.ma".
+
+inline procedural "LambdaDelta-1/lift/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift/fwd.ma".
+
+include "LambdaDelta-2/s/props.ma".
+
+inline procedural "LambdaDelta-1/lift/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift/fwd.ma".
+
+include "LambdaDelta-2/tlt/props.ma".
+
+inline procedural "LambdaDelta-1/lift/tlt.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift1/defs.ma".
+
+include "LambdaDelta-2/lift/fwd.ma".
+
+inline procedural "LambdaDelta-1/lift1/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift/props.ma".
+
+include "LambdaDelta-2/drop1/defs.ma".
+
+inline procedural "LambdaDelta-1/lift1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/A/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/llt/defs.ma".
+
+include "LambdaDelta-2/leq/defs.ma".
+
+inline procedural "LambdaDelta-1/llt/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/G/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/next_plus/defs.ma".
+
+inline procedural "LambdaDelta-1/next_plus/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/nf2/fwd.ma".
+
+include "LambdaDelta-2/arity/subst0.ma".
+
+inline procedural "LambdaDelta-1/nf2/arity.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/nf2/defs.ma".
+
+include "LambdaDelta-2/pr2/clen.ma".
+
+include "LambdaDelta-2/pr2/fwd.ma".
+
+include "LambdaDelta-2/pr0/dec.ma".
+
+include "LambdaDelta-2/C/props.ma".
+
+inline procedural "LambdaDelta-1/nf2/dec.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr2/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/nf2/defs.ma".
+
+include "LambdaDelta-2/pr2/clen.ma".
+
+include "LambdaDelta-2/subst0/dec.ma".
+
+include "LambdaDelta-2/T/props.ma".
+
+inline procedural "LambdaDelta-1/nf2/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/nf2/pr3.ma".
+
+include "LambdaDelta-2/pr3/fwd.ma".
+
+include "LambdaDelta-2/iso/props.ma".
+
+inline procedural "LambdaDelta-1/nf2/iso.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/nf2/props.ma".
+
+include "LambdaDelta-2/drop1/fwd.ma".
+
+inline procedural "LambdaDelta-1/nf2/lift1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/nf2/defs.ma".
+
+include "LambdaDelta-2/pr3/pr3.ma".
+
+inline procedural "LambdaDelta-1/nf2/pr3.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/nf2/defs.ma".
+
+include "LambdaDelta-2/pr2/fwd.ma".
+
+inline procedural "LambdaDelta-1/nf2/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr1/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc1/defs.ma".
+
+include "LambdaDelta-2/pr1/pr1.ma".
+
+inline procedural "LambdaDelta-1/pc1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/arity_props.ma".
+
+include "LambdaDelta-2/nf2/fwd.ma".
+
+inline procedural "LambdaDelta-1/pc3/dec.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/left.ma".
+
+include "LambdaDelta-2/fsubst0/defs.ma".
+
+include "LambdaDelta-2/csubst0/getl.ma".
+
+inline procedural "LambdaDelta-1/pc3/fsubst0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/props.ma".
+
+include "LambdaDelta-2/pr3/fwd.ma".
+
+inline procedural "LambdaDelta-1/pc3/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/props.ma".
+
+inline procedural "LambdaDelta-1/pc3/left.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/defs.ma".
+
+include "LambdaDelta-2/nf2/pr3.ma".
+
+inline procedural "LambdaDelta-1/pc3/nf2.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/defs.ma".
+
+include "LambdaDelta-2/pc1/defs.ma".
+
+include "LambdaDelta-2/pr3/pr1.ma".
+
+inline procedural "LambdaDelta-1/pc3/pc1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/defs.ma".
+
+include "LambdaDelta-2/pr3/pr3.ma".
+
+inline procedural "LambdaDelta-1/pc3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/props.ma".
+
+include "LambdaDelta-2/pr3/subst1.ma".
+
+inline procedural "LambdaDelta-1/pc3/subst1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/props.ma".
+
+include "LambdaDelta-2/wcpr0/getl.ma".
+
+inline procedural "LambdaDelta-1/pc3/wcpr0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/fwd.ma".
+
+include "LambdaDelta-2/subst0/dec.ma".
+
+include "LambdaDelta-2/T/dec.ma".
+
+include "LambdaDelta-2/T/props.ma".
+
+inline procedural "LambdaDelta-1/pr0/dec.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/props.ma".
+
+inline procedural "LambdaDelta-1/pr0/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/fwd.ma".
+
+include "LambdaDelta-2/lift/tlt.ma".
+
+inline procedural "LambdaDelta-1/pr0/pr0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/defs.ma".
+
+include "LambdaDelta-2/subst0/subst0.ma".
+
+inline procedural "LambdaDelta-1/pr0/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/props.ma".
+
+include "LambdaDelta-2/subst1/defs.ma".
+
+inline procedural "LambdaDelta-1/pr0/subst1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr1/props.ma".
+
+include "LambdaDelta-2/pr0/pr0.ma".
+
+inline procedural "LambdaDelta-1/pr1/pr1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr1/defs.ma".
+
+include "LambdaDelta-2/pr0/subst1.ma".
+
+include "LambdaDelta-2/subst1/props.ma".
+
+include "LambdaDelta-2/T/props.ma".
+
+inline procedural "LambdaDelta-1/pr1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr2/props.ma".
+
+include "LambdaDelta-2/clen/getl.ma".
+
+inline procedural "LambdaDelta-1/pr2/clen.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/defs.ma".
+
+include "LambdaDelta-2/getl/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr2/defs.ma".
+
+include "LambdaDelta-2/pr0/fwd.ma".
+
+include "LambdaDelta-2/getl/drop.ma".
+
+include "LambdaDelta-2/getl/clear.ma".
+
+inline procedural "LambdaDelta-1/pr2/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr2/defs.ma".
+
+include "LambdaDelta-2/pr0/pr0.ma".
+
+include "LambdaDelta-2/getl/props.ma".
+
+inline procedural "LambdaDelta-1/pr2/pr2.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr2/defs.ma".
+
+include "LambdaDelta-2/pr0/props.ma".
+
+include "LambdaDelta-2/getl/drop.ma".
+
+include "LambdaDelta-2/getl/clear.ma".
+
+inline procedural "LambdaDelta-1/pr2/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr2/defs.ma".
+
+include "LambdaDelta-2/pr0/subst1.ma".
+
+include "LambdaDelta-2/pr0/fwd.ma".
+
+include "LambdaDelta-2/csubst1/getl.ma".
+
+include "LambdaDelta-2/csubst1/fwd.ma".
+
+include "LambdaDelta-2/subst1/subst1.ma".
+
+include "LambdaDelta-2/getl/drop.ma".
+
+inline procedural "LambdaDelta-1/pr2/subst1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr2/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/props.ma".
+
+include "LambdaDelta-2/pr2/fwd.ma".
+
+inline procedural "LambdaDelta-1/pr3/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/fwd.ma".
+
+include "LambdaDelta-2/iso/props.ma".
+
+include "LambdaDelta-2/tlist/props.ma".
+
+inline procedural "LambdaDelta-1/pr3/iso.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/defs.ma".
+
+include "LambdaDelta-2/pr1/defs.ma".
+
+inline procedural "LambdaDelta-1/pr3/pr1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/props.ma".
+
+include "LambdaDelta-2/pr2/pr2.ma".
+
+inline procedural "LambdaDelta-1/pr3/pr3.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/pr1.ma".
+
+include "LambdaDelta-2/pr2/props.ma".
+
+include "LambdaDelta-2/pr1/props.ma".
+
+inline procedural "LambdaDelta-1/pr3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/defs.ma".
+
+include "LambdaDelta-2/pr2/subst1.ma".
+
+inline procedural "LambdaDelta-1/pr3/subst1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/props.ma".
+
+include "LambdaDelta-2/wcpr0/getl.ma".
+
+inline procedural "LambdaDelta-1/pr3/wcpr0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Base-2/theory.ma".
+include "LambdaDelta-1/definitions.ma".
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/r/defs.ma".
+
+include "LambdaDelta-2/s/defs.ma".
+
+inline procedural "LambdaDelta-1/r/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/s/defs.ma".
+
+inline procedural "LambdaDelta-1/s/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubc/arity.ma".
+
+include "LambdaDelta-2/csubc/getl.ma".
+
+include "LambdaDelta-2/csubc/drop1.ma".
+
+include "LambdaDelta-2/csubc/props.ma".
+
+inline procedural "LambdaDelta-1/sc3/arity.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sn3/defs.ma".
+
+include "LambdaDelta-2/arity/defs.ma".
+
+include "LambdaDelta-2/drop1/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sc3/defs.ma".
+
+include "LambdaDelta-2/sn3/lift1.ma".
+
+include "LambdaDelta-2/nf2/lift1.ma".
+
+include "LambdaDelta-2/csuba/arity.ma".
+
+include "LambdaDelta-2/arity/lift1.ma".
+
+include "LambdaDelta-2/arity/aprem.ma".
+
+include "LambdaDelta-2/llt/props.ma".
+
+include "LambdaDelta-2/drop1/getl.ma".
+
+include "LambdaDelta-2/drop1/props.ma".
+
+include "LambdaDelta-2/lift1/props.ma".
+
+inline procedural "LambdaDelta-1/sc3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr3/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sn3/defs.ma".
+
+include "LambdaDelta-2/pr3/props.ma".
+
+inline procedural "LambdaDelta-1/sn3/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sn3/props.ma".
+
+include "LambdaDelta-2/drop1/fwd.ma".
+
+include "LambdaDelta-2/lift1/fwd.ma".
+
+inline procedural "LambdaDelta-1/sn3/lift1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sn3/defs.ma".
+
+include "LambdaDelta-2/nf2/dec.ma".
+
+include "LambdaDelta-2/nf2/pr3.ma".
+
+inline procedural "LambdaDelta-1/sn3/nf2.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sn3/nf2.ma".
+
+include "LambdaDelta-2/sn3/fwd.ma".
+
+include "LambdaDelta-2/nf2/iso.ma".
+
+include "LambdaDelta-2/pr3/iso.ma".
+
+inline procedural "LambdaDelta-1/sn3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/G/defs.ma".
+
+include "LambdaDelta-2/getl/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sty0/defs.ma".
+
+inline procedural "LambdaDelta-1/sty0/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sty0/defs.ma".
+
+include "LambdaDelta-2/getl/drop.ma".
+
+inline procedural "LambdaDelta-1/sty0/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sty1/props.ma".
+
+include "LambdaDelta-2/cnt/props.ma".
+
+inline procedural "LambdaDelta-1/sty1/cnt.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sty0/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/sty1/defs.ma".
+
+include "LambdaDelta-2/sty0/props.ma".
+
+inline procedural "LambdaDelta-1/sty1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst/defs.ma".
+
+inline procedural "LambdaDelta-1/subst/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst/fwd.ma".
+
+include "LambdaDelta-2/subst0/defs.ma".
+
+include "LambdaDelta-2/lift/props.ma".
+
+inline procedural "LambdaDelta-1/subst/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/defs.ma".
+
+include "LambdaDelta-2/lift/props.ma".
+
+inline procedural "LambdaDelta-1/subst0/dec.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/lift/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/defs.ma".
+
+include "LambdaDelta-2/lift/props.ma".
+
+inline procedural "LambdaDelta-1/subst0/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/fwd.ma".
+
+inline procedural "LambdaDelta-1/subst0/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/props.ma".
+
+inline procedural "LambdaDelta-1/subst0/subst0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/defs.ma".
+
+include "LambdaDelta-2/lift/props.ma".
+
+include "LambdaDelta-2/lift/tlt.ma".
+
+inline procedural "LambdaDelta-1/subst0/tlt.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst1/defs.ma".
+
+include "LambdaDelta-2/subst0/props.ma".
+
+inline procedural "LambdaDelta-1/subst1/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst1/defs.ma".
+
+include "LambdaDelta-2/subst0/props.ma".
+
+inline procedural "LambdaDelta-1/subst1/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst1/fwd.ma".
+
+include "LambdaDelta-2/subst0/subst0.ma".
+
+inline procedural "LambdaDelta-1/subst1/subst1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/subst0/tlt.ma".
+
+include "LambdaDelta-2/subst/props.ma".
+
+include "LambdaDelta-2/sty1/cnt.ma".
+
+include "LambdaDelta-2/ex0/props.ma".
+
+include "LambdaDelta-2/wcpr0/fwd.ma".
+
+include "LambdaDelta-2/pr3/wcpr0.ma".
+
+include "LambdaDelta-2/ex2/props.ma".
+
+include "LambdaDelta-2/ex1/props.ma".
+
+include "LambdaDelta-2/ty3/sty0.ma".
+
+include "LambdaDelta-2/csubt/csuba.ma".
+
+include "LambdaDelta-2/ty3/fwd_nf2.ma".
+
+include "LambdaDelta-2/ty3/nf2.ma".
+
+include "LambdaDelta-2/wf3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/tlist/defs.ma".
+
+inline procedural "LambdaDelta-1/tlist/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/T/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/tlt/defs.ma".
+
+inline procedural "LambdaDelta-1/tlt/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/pr3_props.ma".
+
+include "LambdaDelta-2/arity/pr3.ma".
+
+include "LambdaDelta-2/asucc/fwd.ma".
+
+inline procedural "LambdaDelta-1/ty3/arity.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/arity.ma".
+
+include "LambdaDelta-2/sc3/arity.ma".
+
+inline procedural "LambdaDelta-1/ty3/arity_props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pc3/dec.ma".
+
+include "LambdaDelta-2/getl/flt.ma".
+
+include "LambdaDelta-2/getl/dec.ma".
+
+inline procedural "LambdaDelta-1/ty3/dec.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/G/defs.ma".
+
+include "LambdaDelta-2/pc3/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/props.ma".
+
+include "LambdaDelta-2/pc3/fsubst0.ma".
+
+include "LambdaDelta-2/getl/getl.ma".
+
+inline procedural "LambdaDelta-1/ty3/fsubst0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/defs.ma".
+
+include "LambdaDelta-2/pc3/props.ma".
+
+inline procedural "LambdaDelta-1/ty3/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/arity_props.ma".
+
+include "LambdaDelta-2/pc3/nf2.ma".
+
+include "LambdaDelta-2/nf2/fwd.ma".
+
+inline procedural "LambdaDelta-1/ty3/fwd_nf2.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/arity.ma".
+
+include "LambdaDelta-2/pc3/nf2.ma".
+
+include "LambdaDelta-2/nf2/arity.ma".
+
+inline procedural "LambdaDelta-1/ty3/nf2.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/csubt/ty3.ma".
+
+include "LambdaDelta-2/ty3/subst1.ma".
+
+include "LambdaDelta-2/ty3/fsubst0.ma".
+
+include "LambdaDelta-2/pc3/pc1.ma".
+
+include "LambdaDelta-2/pc3/wcpr0.ma".
+
+include "LambdaDelta-2/pc1/props.ma".
+
+inline procedural "LambdaDelta-1/ty3/pr3.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/pr3.ma".
+
+inline procedural "LambdaDelta-1/ty3/pr3_props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/fwd.ma".
+
+include "LambdaDelta-2/pc3/fwd.ma".
+
+inline procedural "LambdaDelta-1/ty3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/pr3_props.ma".
+
+include "LambdaDelta-2/sty0/fwd.ma".
+
+inline procedural "LambdaDelta-1/ty3/sty0.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/props.ma".
+
+include "LambdaDelta-2/pc3/subst1.ma".
+
+include "LambdaDelta-2/getl/getl.ma".
+
+inline procedural "LambdaDelta-1/ty3/subst1.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/pr0/defs.ma".
+
+include "LambdaDelta-2/C/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/wcpr0/defs.ma".
+
+inline procedural "LambdaDelta-1/wcpr0/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/wcpr0/defs.ma".
+
+include "LambdaDelta-2/getl/props.ma".
+
+inline procedural "LambdaDelta-1/wcpr0/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/wf3/fwd.ma".
+
+inline procedural "LambdaDelta-1/wf3/clear.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/ty3/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/wf3/defs.ma".
+
+inline procedural "LambdaDelta-1/wf3/fwd.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/wf3/clear.ma".
+
+include "LambdaDelta-2/ty3/dec.ma".
+
+inline procedural "LambdaDelta-1/wf3/getl.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/wf3/ty3.ma".
+
+include "LambdaDelta-2/app/defs.ma".
+
+inline procedural "LambdaDelta-1/wf3/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-2/wf3/getl.ma".
+
+inline procedural "LambdaDelta-1/wf3/ty3.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/preamble.ma".
+
+definition blt:
+ nat \to (nat \to bool)
+\def
+ let rec blt (m: nat) (n: nat) on n: bool \def (match n with [O \Rightarrow
+false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S m0)
+\Rightarrow (blt m0 n0)])]) in blt.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/blt/defs.ma".
+
+theorem lt_blt:
+ \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
+(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0
+\def (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat
+n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq
+nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in
+nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) |
+(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind
+nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop)
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind
+((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal
+nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to
+(eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
+nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S
+n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n))
+\to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m
+n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1)))))
+y)))) x).
+
+theorem le_bge:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
+(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O
+y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y:
+nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind
+(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false)))
+(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le return (\lambda (n0:
+nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n))
+false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
+(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
+(False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow
+(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
+| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool
+(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0:
+nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n))
+false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y))))
+x).
+
+theorem blt_lt:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
+y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O)
+true)).(let H0 \def (match H in eq return (\lambda (b: bool).(\lambda (_: (eq
+? ? b)).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow
+(\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O)
+(\lambda (e: bool).(match e in bool return (\lambda (_: bool).Prop) with
+[true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind
+(lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda
+(H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda
+(y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt
+n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S
+O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq
+bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)
+\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H
+n0 H1))))) y)))) x).
+
+theorem bge_le:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
+y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O)
+false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq
+bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda
+(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0:
+(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq return (\lambda
+(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O))))
+with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n))
+false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e
+in bool return (\lambda (_: bool).Prop) with [true \Rightarrow True | false
+\Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1
+(refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0
+(S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n))
+false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0
+H1))))))) y)))) x).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/types/defs.ma".
+
+include "Base-1/blt/defs.ma".
+
+include "Base-1/plist/defs.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/preamble.ma".
+
+theorem nat_dec:
+ \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat
+n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2:
+nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall
+(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P:
+Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n)
+((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq
+nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S
+n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match
+ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
+\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n:
+nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to
+(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or
+(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror
+(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0:
+(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda
+(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
+False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda
+(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall
+(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
+Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P:
+Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0
+(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P:
+Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S
+n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat
+(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat
+(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P:
+Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to
+(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P:
+Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
+(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n)
+(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3)
+\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0
+(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0:
+Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2))))
+n1).
+
+theorem simpl_plus_r:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n)
+(plus p n)) \to (eq nat m p))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
+(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda
+(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
+nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n
+p)) (plus m n) H) (plus n m) (plus_sym n m)))))).
+
+theorem minus_Sx_Sy:
+ \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
+
+theorem minus_plus_r:
+ \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
+\def
+ \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
+nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
+
+theorem plus_permute_2_in_3:
+ \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
+y) z) (plus (plus x z) y))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x
+(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat
+(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind
+nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
+(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z
+y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
+
+theorem plus_permute_2_in_3_assoc:
+ \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
+h) k) (plus n (plus k h)))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus
+(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r
+nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
+(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k
+h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
+
+theorem plus_O:
+ \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
+x O) (eq nat y O))))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus
+n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda
+(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O)
+H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O)
+\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq
+nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0:
+nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq
+nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n)
+y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in
+nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y
+O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
+
+theorem minus_Sx_SO:
+ \forall (x: nat).(eq nat (minus (S x) (S O)) x)
+\def
+ \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal
+nat x) (minus x O) (minus_n_O x)).
+
+theorem eq_nat_dec:
+ \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq
+nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or
+(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O)
+(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq
+nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j))
+(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq
+nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat
+(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S
+n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or
+(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq
+nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda
+(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S
+n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
+(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H
+n0)))) j)))) i).
+
+theorem neq_eq_e:
+ \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j))
+\to P)) \to ((((eq nat i j) \to P)) \to P))))
+\def
+ \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not
+(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def
+(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
+
+theorem le_false:
+ \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S
+n) m) \to P))))
+\def
+ \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P:
+Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P:
+Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match
+H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to
+P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
+(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
+(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0)
+O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal
+nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P:
+Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind
+(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n))
+\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S
+O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda
+(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2:
+(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e
+in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow
+(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda
+(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
+False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P)
+H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_:
+((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda
+(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S
+n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
+
+theorem le_Sx_x:
+ \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
+\def
+ \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
+le_Sn_n in (False_ind P (H0 x H))))).
+
+theorem le_n_pred:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
+(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
+nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
+(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
+
+theorem minus_le:
+ \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n
+y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall
+(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0:
+nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda
+(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)])
+(S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
+
+theorem le_plus_minus_sym:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n)
+n))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
+(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
+(plus (minus m n) n) (plus_sym (minus m n) n)))).
+
+theorem le_minus_minus:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
+\to (le (minus y x) (minus z x))))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z:
+nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x)
+(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
+z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
+(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
+
+theorem le_minus_plus:
+ \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat
+(minus (plus x y) z) (plus (minus x z) y)))))
+\def
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to
+(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y))))))
+(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return
+(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y:
+nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n
+\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n:
+nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))
+(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O)
+(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq
+nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y:
+nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O
+m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))])
+in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x:
+nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus
+(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S
+z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
+(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def
+(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
+\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with
+[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat
+(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with
+[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq
+nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1)
+\Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m)
+(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S
+z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3))
+H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S
+z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
+(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n
+(le_S_n z0 n H1) y))))) x)))) z).
+
+theorem le_minus:
+ \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to
+(le x (minus z y)))))
+\def
+ \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus
+x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z
+y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x
+y))))).
+
+theorem le_trans_plus_r:
+ \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to
+(le y z))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus
+x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
+
+theorem lt_x_O:
+ \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
+\def
+ \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
+(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
+ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
+\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))).
+
+theorem le_gen_S:
+ \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
+nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
+\def
+ \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def
+(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x)
+\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m
+n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
+(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0)))
+(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S
+m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x
+H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat
+(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq
+nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m)
+m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
+nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2))))
+x H1 H0))]) in (H0 (refl_equal nat x))))).
+
+theorem lt_x_plus_x_Sy:
+ \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
+nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
+(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
+
+theorem simpl_lt_plus_r:
+ \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
+p)) \to (lt n m))))
+\def
+ \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus
+n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p)
+(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let
+H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
+(plus p m) (plus_sym m p)) in H1)))))).
+
+theorem minus_x_Sy:
+ \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
+(minus x (S y))))))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
+(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H:
+(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_:
+(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with
+[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S
+y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
+(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda
+(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
+| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat
+(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O)))))
+(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat
+(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
+nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))
+(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S
+n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0:
+nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S
+n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0)
+n H1) in (H n0 H2))))) y)))) x).
+
+theorem lt_plus_minus:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus
+y (S x)))))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S
+x) y H))).
+
+theorem lt_plus_minus_r:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y
+(S x)) x)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
+(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
+y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
+
+theorem minus_x_SO:
+ \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
+\def
+ \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n:
+nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal
+nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
+
+theorem le_x_pred_y:
+ \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
+\def
+ \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to
+(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def
+(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
+\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let
+H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0)
+in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat
+(S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
+(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt
+x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S
+n))).(le_S_n x n H0))))) y).
+
+theorem lt_le_minus:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
+O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
+(plus_sym x (S O)))))).
+
+theorem lt_le_e:
+ \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
+\to ((((le d n) \to P)) \to P))))
+\def
+ \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n
+d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in
+(or_ind (le d n) (lt n d) P H0 H H1)))))).
+
+theorem lt_eq_e:
+ \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
+\to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
+y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x
+y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
+
+theorem lt_eq_gt_e:
+ \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
+\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
+y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
+\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda
+(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
+
+theorem lt_gen_xS:
+ \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2
+nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S
+n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m)))
+(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S
+n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m)))
+(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda
+(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda
+(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0:
+nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat
+(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0)))
+(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt
+m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
+
+theorem le_lt_false:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P:
+Prop).P))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt
+y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
+
+theorem lt_neq:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
+nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
+(lt_n_n y H1))))).
+
+theorem arith0:
+ \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
+\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
+\def
+ \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le
+(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1))
+h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2
+(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat
+(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat
+(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus
+(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1)
+(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2
+d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1)
+(minus_plus h2 (plus d2 h1))))))).
+
+theorem O_minus:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
+(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O
+y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y:
+nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind
+(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
+x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0)
+O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0
+n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S
+x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
+| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O)
+H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n)
+\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0
+l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y))))
+x).
+
+theorem minus_minus:
+ \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y)
+\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
+\def
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y:
+nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq
+nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O
+x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y
+O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n
+(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O)
+(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda
+(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to
+((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x
+y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le
+(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0)))
+\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda
+(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S
+z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le
+z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S
+x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
+| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4)))))
+(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y:
+nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus
+y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n:
+nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S
+z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S
+x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0))
+(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda
+(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O)
+(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0
+x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0))))))
+(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to
+((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0)
+y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S
+y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S
+z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0)
+H1))))))) y)))) x)))) z).
+
+theorem plus_plus:
+ \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
+nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z
+x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
+\def
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2:
+nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq
+nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2)
+(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1:
+nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2
+O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat
+(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O
+(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def
+(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus
+O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1))))))))
+(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2:
+nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to
+((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus
+x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n:
+nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0))
+\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S
+z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2:
+nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
+(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus
+(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda
+(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O
+(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y
+\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
+nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq
+nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0))
+in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2)
+H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2:
+nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus
+(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2))
+\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
+nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda
+(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O
+x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S
+y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0
+(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n:
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus
+(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1))
+(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda
+(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus
+z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1))
+(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1))))))))))))
+x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1:
+nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat
+(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2
+y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n:
+nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S
+z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))
+\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda
+(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S
+z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let
+H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O)
+(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2)
+y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O
+z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2
+z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus
+x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def
+(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0)
+\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1)))))
+H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0)
+H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall
+(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus
+z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow
+(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4
+y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S
+z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0
+x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
+y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3))))
+x1)))) z).
+
+theorem le_S_minus:
+ \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
+(le d (S (minus n h))))))
+\def
+ \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
+d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
+\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
+(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S
+d (minus n h) (le_minus d n h H))))))).
+
+theorem lt_x_pred_y:
+ \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
+n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
+(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
+(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x)
+n H0)))))) y)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/preamble.ma".
+
+theorem insert_eq:
+ \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G:
+((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) \to
+((P x) \to (G x))))))
+\def
+ \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G:
+((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to (G
+y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))).
+
+theorem unintro:
+ \forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x:
+A).(P x))) \to (P a))))
+\def
+ \lambda (A: Set).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda (H:
+((\forall (x: A).(P x)))).(H a)))).
+
+theorem xinduction:
+ \forall (A: Set).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall (x:
+A).((eq A t x) \to (P x)))) \to (P t))))
+\def
+ \lambda (A: Set).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda (H:
+((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/preamble.ma".
+
+inductive PList: Set \def
+| PNil: PList
+| PCons: nat \to (nat \to (PList \to PList)).
+
+definition PConsTail:
+ PList \to (nat \to (nat \to PList))
+\def
+ let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def
+(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow
+(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0
+h0 d0))]))) in PConsTail.
+
+definition Ss:
+ PList \to PList
+\def
+ let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow
+PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss.
+
+definition papp:
+ PList \to (PList \to PList)
+\def
+ let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b:
+PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons
+h d (papp a0 b))])) in papp.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/plist/defs.ma".
+
+theorem papp_ss:
+ \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss
+is2)) (Ss (papp is1 is2))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2:
+PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2:
+PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0:
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp
+(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList
+(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n
+(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p
+is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Legacy-1/theory.ma".
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/theory.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/ext/tactics.ma".
+
+include "Base-1/ext/arith.ma".
+
+include "Base-1/types/props.ma".
+
+include "Base-1/blt/props.ma".
+
+include "Base-1/plist/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/preamble.ma".
+
+inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
+| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))).
+
+inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
+| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))).
+
+inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop
+\def
+| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3
+P4))))).
+
+inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
+| or3_intro0: P0 \to (or3 P0 P1 P2)
+| or3_intro1: P1 \to (or3 P0 P1 P2)
+| or3_intro2: P2 \to (or3 P0 P1 P2).
+
+inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
+| or4_intro0: P0 \to (or4 P0 P1 P2 P3)
+| or4_intro1: P1 \to (or4 P0 P1 P2 P3)
+| or4_intro2: P2 \to (or4 P0 P1 P2 P3)
+| or4_intro3: P3 \to (or4 P0 P1 P2 P3).
+
+inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop
+\def
+| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4).
+
+inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to
+Prop): Prop \def
+| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0
+P0 P1 P2)))).
+
+inductive ex4 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop)
+(P3: A0 \to Prop): Prop \def
+| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0)
+\to (ex4 A0 P0 P1 P2 P3))))).
+
+inductive ex_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)): Prop \def
+| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1
+P0))).
+
+inductive ex2_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to
+(A1 \to Prop)): Prop \def
+| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1)
+\to (ex2_2 A0 A1 P0 P1)))).
+
+inductive ex3_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to
+(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def
+| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1)
+\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))).
+
+inductive ex4_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to
+(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): Prop
+\def
+| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1)
+\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))).
+
+inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
+Prop))): Prop \def
+| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1
+x2) \to (ex_3 A0 A1 A2 P0)))).
+
+inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
+Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def
+| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
+x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))).
+
+inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
+Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to
+Prop))): Prop \def
+| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
+x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1
+P2)))))).
+
+inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
+Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to
+Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def
+| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
+x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0
+A1 A2 P0 P1 P2 P3))))))).
+
+inductive ex5_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to
+Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to
+Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to
+Prop))): Prop \def
+| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0
+x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0
+x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))).
+
+inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to
+(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0
+\to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def
+| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
+(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to
+(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))).
+
+inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to
+(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0
+\to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 \to (A3 \to
+Prop)))): Prop \def
+| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
+(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to
+((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))).
+
+inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to
+(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to
+(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3:
+A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def
+| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
+(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to
+((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1
+P2 P3))))))))).
+
+inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to
+(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to
+(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3:
+A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: A0 \to (A1 \to (A2 \to
+(A3 \to (A4 \to Prop))))): Prop \def
+| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
+(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to
+((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to
+(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))).
+
+inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set)
+(P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1: A0 \to
+(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P2: A0 \to (A1 \to (A2
+\to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to
+(A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5
+\to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to
+Prop)))))): Prop \def
+| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
+(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1
+x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5)
+\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2
+A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))).
+
+inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set)
+(A6: Set) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
+Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
+Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
+Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
+Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
+Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to
+Prop))))))): Prop \def
+| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall
+(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2
+x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6)
+\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1
+x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4
+P5))))))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-1/types/defs.ma".
+
+theorem ex2_sym:
+ \forall (A: Set).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to
+Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A
+(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))))))
+\def
+ \lambda (A: Set).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to
+Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q
+x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A
+(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0:
+(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda
+(x0: A).(P x0)) x H1 H0)))) H)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/preamble.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/blt/defs.ma".
+
+inline procedural "Base-1/blt/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/preamble.ma".
+
+inline procedural "Base-1/ext/arith.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/preamble.ma".
+
+inline procedural "Base-1/ext/tactics.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/preamble.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/plist/defs.ma".
+
+inline procedural "Base-1/plist/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Base-1/definitions.ma".
+include "Legacy-2/theory.ma".
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/ext/tactics.ma".
+
+include "Base-2/ext/arith.ma".
+
+include "Base-2/types/props.ma".
+
+include "Base-2/blt/props.ma".
+
+include "Base-2/plist/props.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/preamble.ma".
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Base-2/types/defs.ma".
+
+inline procedural "Base-1/types/props.ma".
+
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/preamble.ma".
-
-inductive A: Set \def
-| ASort: nat \to (nat \to A)
-| AHead: A \to (A \to A).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-inductive C: Set \def
-| CSort: nat \to C
-| CHead: C \to (K \to (T \to C)).
-
-definition cweight:
- C \to nat
-\def
- let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O
-| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight.
-
-definition clt:
- C \to (C \to Prop)
-\def
- \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))).
-
-definition cle:
- C \to (C \to Prop)
-\def
- \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))).
-
-definition CTail:
- K \to (T \to (C \to C))
-\def
- let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n)
-\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k
-t d) h u)]) in CTail.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-include "LambdaDelta-1/T/props.ma".
-
-theorem clt_cong:
- \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
-T).(clt (CHead c k t) (CHead d k t))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
-d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
-(tweight t) H))))).
-
-theorem clt_head:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
-c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u))))
-(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
-(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
-
-theorem clt_wf__q_ind:
- \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to
-Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0
-c))))) P n))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
-n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight
-c)))))).
-
-theorem clt_wf_ind:
- \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
-\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
-Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
-(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind
-(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0:
-C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
-\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
-(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
-(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P
-c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt
-(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight
-d))))))))))))) c)))).
-
-theorem chead_ctail:
- \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
-K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3
-K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t)
-(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k:
-K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
-(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead
-(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall
-(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
-(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def
-H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
-(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d:
-C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d))))))
-(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead
-c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1:
-C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead
-c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d:
-C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
-(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0
-k t) H1))))) H0))))))))) c).
-
-theorem clt_thead:
- \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt
-c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
-C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t:
-T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
-
-theorem c_tail_ind:
- \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
-(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t
-c))))))) \to (\forall (c: C).(P c))))
-\def
- \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort
-n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t:
-T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P
-c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1)
-\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d:
-C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_:
-((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to
-(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind
-K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t)
-(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2
-x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def
-(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P
-d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2))
-(CHead c1 k t) H4))))) H3)))))))) c0)) c)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/preamble.ma".
-
-record G : Set \def {
- next: (nat \to nat);
- next_lt: (\forall (n: nat).(lt n (next n)))
-}.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-theorem terms_props__bind_dec:
- \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall
-(P: Prop).P))))
-\def
- \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq
-B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b:
-B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl
-(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B
-Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P:
-Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def
-(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
-with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow
-False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B
-Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda
-(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
-Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda
-(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall
-(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall
-(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def
-(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
-with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
-False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B
-Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B
-Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
-Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee:
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
-Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P
-H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B
-Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void
-Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P:
-Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
-Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B
-Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
-Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee:
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
-Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P
-H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P:
-Prop).P)) (refl_equal B Void)) b2)) b1).
-
-theorem bind_dec_not:
- \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
-\def
- \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2)
-in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P:
-Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
-b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
-(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1
-b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
-
-theorem terms_props__flat_dec:
- \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall
-(P: Prop).P))))
-\def
- \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq
-F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f:
-F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl
-(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F
-Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P:
-Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def
-(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop)
-with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in
-(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F
-Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast
-Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast
-Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match
-ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast
-\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast
-Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2))
-f1).
-
-theorem terms_props__kind_dec:
- \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall
-(P: Prop).P))))
-\def
- \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq
-K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind
-(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P:
-Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in
-(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P:
-Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to
-(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1:
-B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P:
-Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to
-(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B
-b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq
-K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b)
-(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e:
-K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
-(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B
-b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in
-(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b)
-(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H:
-(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b)
-(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind
-P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or
-(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda
-(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to
-(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P:
-Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def
-(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F
-f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat
-f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind
-F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1))
-\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat
-f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0))
-(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K
-(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P))
-(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def
-(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with
-[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1)
-in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall
-(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1).
-
-theorem term_dec:
- \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall
-(P: Prop).P))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq
-T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2:
-T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to
-(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in
-(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
-Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to
-(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda
-(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to
-(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort
-n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0))
-(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T
-(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P))
-(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _)
-\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0
-(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in
-(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T
-(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P))
-(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0))))))
-(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T
-(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or
-(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P:
-Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n)
-(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n)
-(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n:
-nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T
-(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T
-(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P))
-(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def
-(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0))))))
-(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind
-(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n)
-(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda
-(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n)
-(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P))))
-(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to
-(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq
-nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0))
-((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T
-(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat
-(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n]))
-(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1:
-nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal
-nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T
-(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0:
-T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P:
-Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n)
-(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n)
-(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t
-t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall
-(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P:
-Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t
-t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n:
-nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort
-n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort
-n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T
-(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P:
-Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P:
-Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in
-(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq
-T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P:
-Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq
-T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in
-(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P:
-Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0)
-(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let
-H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
-(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t
-(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t
-t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in
-(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P:
-Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0)
-(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let
-H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
-(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0
-(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t
-t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def
-(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq
-K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0))
-((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda
-(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0)
-(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P:
-Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t
-t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0)))
-k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror
-(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0))
-\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t
-t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
-t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to
-(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9)))
-t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror
-(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4))
-\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t
-t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
-t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
-| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in
-(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq
-T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r
-T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5)
-\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P)))))
-H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P:
-Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k
-t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead
-k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
-(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t
-| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0)
-(THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3
-t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def
-(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k
-t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r
-T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in
-(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5)
-((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12
-(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
-
-theorem binder_dec:
- \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
-T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall
-(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b:
-B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))
-(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w
-u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T
-(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind
-b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n)
-(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
-(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w
-u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n:
-nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
-T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
-T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P:
-Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq
-T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T
-(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0)))))))))
-(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T
-(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
-u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead
-(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3
-B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind
-b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
-(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda
-(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b)
-w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0
-t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b:
-B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda
-(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0:
-B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to
-(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T
-(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w
-u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
-(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T
-(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1)
-(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u:
-T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P:
-Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u:
-T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal
-T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda
-(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0
-(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u:
-T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda
-(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda
-(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
-T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P:
-Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w:
-T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))))))
-(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1)
-(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
-(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead
-(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0
-t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
-\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1)
-in (False_ind P H2))))))))))))) k)) t).
-
-theorem abst_dec:
- \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead
-(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to
-(\forall (P: Prop).P)))))
-\def
- \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda
-(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead
-(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda
-(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v
-t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall
-(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind
-Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n:
-nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead
-(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t))
-\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n)
-(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0:
-T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind
-Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_:
-((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v
-t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P:
-Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind
-Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind
-Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t
-t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead
-(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind
-Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1:
-T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T
-(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let
-H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t
-v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind
-Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind
-Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda
-(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq
-T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq
-T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P:
-Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0)
-(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0)
-(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda
-(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0
-(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v)
-\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead
-(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead
-(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))
-(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind
-Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t |
-(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst)
-t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind
-Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8
-P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall
-(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead
-(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind
-Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T
-(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _)
-\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
-\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_:
-(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4)))))))
-H1))))))))) u).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/preamble.ma".
-
-inductive B: Set \def
-| Abbr: B
-| Abst: B
-| Void: B.
-
-inductive F: Set \def
-| Appl: F
-| Cast: F.
-
-inductive K: Set \def
-| Bind: B \to K
-| Flat: F \to K.
-
-inductive T: Set \def
-| TSort: nat \to T
-| TLRef: nat \to T
-| THead: K \to (T \to (T \to T)).
-
-definition tweight:
- T \to nat
-\def
- let rec tweight (t: T) on t: nat \def (match t with [(TSort _) \Rightarrow
-(S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus
-(tweight u) (tweight t0)))]) in tweight.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-theorem not_abbr_abst:
- not (eq B Abbr Abst)
-\def
- \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee:
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True |
-Abst \Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind
-False H0)).
-
-theorem not_void_abst:
- not (eq B Void Abst)
-\def
- \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee:
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
-Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind
-False H0)).
-
-theorem not_abbr_void:
- not (eq B Abbr Void)
-\def
- \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee:
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True |
-Abst \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind
-False H0)).
-
-theorem not_abst_void:
- not (eq B Abst Void)
-\def
- \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee:
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
-Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind
-False H0)).
-
-theorem thead_x_y_y:
- \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to
-(\forall (P: Prop).P))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq
-T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda
-(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def
-(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in
-(False_ind P H0))))) (\lambda (n: nat).(\lambda (H: (eq T (THead k v (TLRef
-n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (TLRef
-n)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0:
-K).(\lambda (t0: T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P:
-Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to
-(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1))
-(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
-(TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (THead
-k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v |
-(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) (THead k v (THead
-k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead
-k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2)
-\Rightarrow t2])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in
-(\lambda (H5: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T
-v (\lambda (t2: T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0)))
-H0 t0 H5) in (let H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0
-t1) t1) \to (\forall (P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3))
-H2))))))))) t))).
-
-theorem tweight_lt:
- \forall (t: T).(lt O (tweight t))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_:
-nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda
-(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O
-(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S
-O) (tweight t0) (tweight t1) H))))))) t).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/asucc/defs.ma".
-
-definition aplus:
- G \to (A \to (nat \to A))
-\def
- let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O
-\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/aplus/defs.ma".
-
-include "LambdaDelta-1/next_plus/props.ma".
-
-theorem aplus_reg_r:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
-(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A
-(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda
-(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h:
-nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2
-(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n
-h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n
-h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))).
-
-theorem aplus_assoc:
- \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A
-(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n:
-nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n
-h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n:
-nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus
-g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A
-(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))))
-(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g
-(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O
-n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n))
-n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda
-(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g
-(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n))
-n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0))
-(plus_n_Sm n n0)))) h2)))) h1))).
-
-theorem aplus_asucc:
- \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a)
-h) (asucc g (aplus g a h)))))
-\def
- \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a
-(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h))))
-(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h)
-(aplus_assoc g a (S O) h)))).
-
-theorem aplus_sort_O_S_simpl:
- \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O
-n) (S k)) (aplus g (ASort O (next g n)) k))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc
-g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k)))
-(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n)
-k)) (aplus_asucc g k (ASort O n))))).
-
-theorem aplus_sort_S_S_simpl:
- \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A
-(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind
-A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g
-(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g
-(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
-
-theorem aplus_asort_O_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O
-n) h) (ASort O (next_plus g n h)))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0:
-nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda
-(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall
-(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0
-n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n)
-(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat
-(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next
-g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n))
-(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n
-(ASort O n0)))))) h)).
-
-theorem aplus_asort_le_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h
-k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k:
-nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort
-(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O
-k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n)))
-(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0:
-nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A
-(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k:
-nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A
-(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda
-(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat
-O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n)
-h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S
-x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus
-g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0))))
-(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A
-(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda
-(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort
-(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n
-n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g
-h0 (ASort (S n) n0))))))) k)))) h)).
-
-theorem aplus_asort_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A
-(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n:
-nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus
-g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k))
-(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h)
-(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k)
-(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus
-h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a
-(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O
-(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k
-h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A
-(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k)))))
-(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h
-(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k)
-(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus
-h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h
-(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort
-(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n
-(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h)
-n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h)
-(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h)
-(aplus_asort_le_simpl g h k n H))))))).
-
-theorem aplus_ahead_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A
-(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1:
-A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2
-n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2))))
-(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A
-(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1:
-A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda
-(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g
-(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n)
-(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n
-a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2)))))))
-h)).
-
-theorem aplus_asucc_false:
- \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a)
-h) a) \to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h:
-nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P))))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A
-(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0)
-\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind
-(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O
-(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P))
-(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1
-\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0
-(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
-(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat
-(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1)
-\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1:
-nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0)
-\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0)
-(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
-(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1:
-nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in
-(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g
-n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n
-(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda
-(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort
-O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to
-P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let
-H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort
-(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
-(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e:
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow
-n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat)
-\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow
-(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in
-minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S
-n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _)
-\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def
-(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0
-n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus
-g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1
-h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2
-n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0:
-A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to
-(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h:
-nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P:
-Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc
-g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g
-(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1
-(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1)))
-in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
-(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n:
-nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc
-g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow
-a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3
-P)))))))))) a)).
-
-theorem aplus_inj:
- \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A
-(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2:
-nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
-h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A
-(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_:
-(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a:
-A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0:
-(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g
-a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g
-n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq
-nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2:
-nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
-h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq
-A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a:
-A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A
-(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a)
-n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O)))))
-(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a
-n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1:
-(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def
-(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus
-g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def
-(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g
-a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat
-nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-definition cbk:
- C \to nat
-\def
- let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m |
-(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk.
-
-definition app1:
- C \to (T \to T)
-\def
- let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with
-[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u
-t))])) in app1.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/A/defs.ma".
-
-inductive aprem: nat \to (A \to (A \to Prop)) \def
-| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
-| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2
-a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/aprem/defs.ma".
-
-theorem aprem_gen_sort:
- \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem
-i (ASort h n) x) \to False))))
-\def
- \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n:
-nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda
-(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0:
-(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_:
-A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2:
-A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A
-(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop)
-with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
-(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a:
-A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2
-(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2)
-(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee
-in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
-(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False
-H4))))))))) i y x H0))) H))))).
-
-theorem aprem_gen_head_O:
- \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x)
-\to (eq A x a1))))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O
-(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x))
-(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y
-x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq
-A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem
-y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq
-nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0:
-A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0
-a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _)
-\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3)
-in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a:
-A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i
-O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda
-(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let
-H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
-with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3
-a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e
-in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _
-a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A
-a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A
-a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0
-(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i)
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a
-a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
-
-theorem aprem_gen_head_S:
- \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem
-(S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda
-(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a:
-A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y:
-A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n:
-nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2
-x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n:
-nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead
-a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda
-(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let
-H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
-with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3)
-(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a)
-\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0
-a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat
-O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind
-(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a:
-A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat
-i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3:
-A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0)
-(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _)
-\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2)
-H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4:
-A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3
-a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2
-H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat
-return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n]))
-(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n
-(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let
-H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in
-H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/aprem/fwd.ma".
-
-include "LambdaDelta-1/leq/defs.ma".
-
-theorem aprem_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
-(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g
-b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall
-(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
-(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda
-(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g
-(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2:
-A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2
-i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g
-b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda
-(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall
-(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g
-b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda
-(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall
-(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
-(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda
-(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead
-a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem
-n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y
-\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A
-(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))
-(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead
-a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda
-(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2))
-(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0)
-(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let
-H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1:
-A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1:
-A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda
-(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2
-A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0
-a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2
-H)))).
-
-theorem aprem_asucc:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i
-a1 a2) \to (aprem i (asucc g a1) a2)))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda
-(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda
-(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3:
-A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda
-(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0)
-a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2
-H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/arity/props.ma".
-
-include "LambdaDelta-1/arity/cimp.ma".
-
-include "LambdaDelta-1/aprem/props.ma".
-
-theorem arity_aprem:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c))))
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
-b)))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0:
-A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
-b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda
-(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b
-i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d:
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d:
-C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1))))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_:
-(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem
-i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b:
-A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def
-H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0:
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop
-(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0
-\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def
-H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda
-(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0:
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
-(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop
-(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus
-i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2)
-H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst)
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2:
-((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T
-nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
-d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
-(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem
-i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T
-nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
-d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
-(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6:
-(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2)
-x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1:
-C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1
-x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop
-(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_:
-nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop
-(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x
-x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8
-(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7))))))))
-H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
-(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
-a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \to
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
-i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d
-u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
-(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i:
-nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d:
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b)
-u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
-(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5:
-(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind
-C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O
-d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
-nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda
-(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0
-(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9
-\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S
-b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C
-T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
-(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g
-a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1)
-b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
-(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
-nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3:
-((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead
-c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
-nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b:
-A).(\lambda (H4: (aprem i (AHead a1 a2) b)).(nat_ind (\lambda (n:
-nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda
-(_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H5:
-(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in
-(eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a0))))))) (ex2_3_intro C T
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
-(asucc g a1))))) c0 u O (drop_refl c0) H0) b H_y))) (\lambda (i0:
-nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0))))
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
-b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def
-(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6
-\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d:
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C
-T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j)
-O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
-(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
-nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst)
-u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0))))
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
-b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i
-H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
-(_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b:
-A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3:
-((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
-(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem
-i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C
-T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j))
-O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
-(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda
-(H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i
-x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda
-(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))))
-(\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n)
-c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0
-(CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda
-(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))
-(\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2))
-O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2))
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C
-T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
-(asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8)))))
-(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g
-d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k:
-K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1)
-c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda
-(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i
-x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10:
-(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r
-(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0:
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))
-(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n:
-nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d
-c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda
-(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r
-(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g
-(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in
-(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop
-(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_:
-nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda
-(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0:
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop
-(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g
-b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9
-(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7))))))
-H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda
-(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b:
-A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0:
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall
-(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i:
-nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4)
-in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
-b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6:
-(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g
-b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
-nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda
-(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5))))))))))))))
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
-t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
-i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d
-u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1
-a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x
-\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A
-(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
-c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
-g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i
-a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
-x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
-(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
-nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0
-x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_:
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g
-x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7))))))
-H4))))))))))))) c t a H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/arity/defs.ma".
-
-include "LambdaDelta-1/cimp/props.ma".
-
-theorem arity_cimp_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
-t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
-A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
-C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g
-c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
-A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d
-c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c
-c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i)
-a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let
-H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in
-(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead
-d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7:
-(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def
-(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead
-x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind
-Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _)
-\Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono
-c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11
-\def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9
-x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x
-H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6)))))
-H4))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
-A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2:
-C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda
-(H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in
-(ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2
-(TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst)
-u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def
-H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0:
-C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst)
-u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i
-c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i
-H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x |
-(CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind
-Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u)
-H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0
-(Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0:
-C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12)))))))))
-H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b
-Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
-g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
-(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b)
-u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c
-c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u)
-(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
-((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
-a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to
-(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c
-c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
-(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
-(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2:
-A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2:
-C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2:
-C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2
-H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
-(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to
-(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
-a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0
-a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1
-c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
-(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
-C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2:
-(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2
-t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/leq/defs.ma".
-
-include "LambdaDelta-1/getl/defs.ma".
-
-inductive arity (g: G): C \to (T \to (A \to Prop)) \def
-| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O
-n)))
-| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a)
-\to (arity g c (TLRef i) a)))))))
-| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u
-(asucc g a)) \to (arity g c (TLRef i) a)))))))
-| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c:
-C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t:
-T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c
-(THead (Bind b) u t) a2)))))))))
-| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
-(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind
-Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
-| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
-a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to
-(arity g c (THead (Flat Appl) u t) a2)))))))
-| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u
-(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat
-Cast) u t) a))))))
-| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t
-a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/arity/defs.ma".
-
-include "LambdaDelta-1/leq/asucc.ma".
-
-include "LambdaDelta-1/getl/drop.ma".
-
-theorem arity_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c
-(TSort n) a) \to (leq g a (ASort O n))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda
-(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g
-c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0:
-(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0:
-A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_:
-C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1:
-nat).(leq g (ASort O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2)))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(_: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity
-g d u a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O
-n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T
-(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n))
-H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
-A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n))
-\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort
-n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
-(False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_:
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq
-g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
-(CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2
-(ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7
-\def (eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in
-(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda
-(_: (((eq T u (TSort n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t
-a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda
-(H5: (eq T (THead (Bind Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead
-(Bind Abst) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TSort n) H5) in (False_ind (leq g (AHead a1 a2)
-(ASort O n)) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq
-g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
-c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TSort n)) \to (leq g (AHead a1
-a2) (ASort O n))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TSort
-n))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
-H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0:
-C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g
-a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O
-n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t
-(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat
-Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n)) H6)))))))))))
-(\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t
-a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O
-n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t
-(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in
-(let H6 \def (eq_ind T t (\lambda (t0: T).((eq T t0 (TSort n)) \to (leq g a1
-(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t0:
-T).(arity g c0 t0 a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1
-a2 H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0)))
-H))))).
-
-theorem arity_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c
-(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a))))
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst)
-u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda
-(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g
-c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl
-i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind
-Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))
-(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0:
-C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0:
-C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C
-T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (ASort O n)))))))
-H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0:
-nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0:
-A).(\lambda (H2: (arity g d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or
-(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr)
-u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g
-a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal
-T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort
-_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
-(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n:
-nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda
-(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
-(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6 H2)))))))))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda
-(H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2:
-(arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2
-C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g
-a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal
-T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort
-_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
-(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n:
-nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda
-(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
-(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6
-H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
-(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
-a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
-C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
-(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead
-c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
-(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda
-(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind
-b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
-c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
-(asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
-(a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u
-(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
-(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
-(asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
-(CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
-(asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
-(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i))
-\to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind
-Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity
-g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0
-(Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst)
-u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) (\lambda (c0:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda
-(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
-c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
-(asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g
-c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T
-(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1
-a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let H6
-\def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
-(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
-d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2))))
-(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst)
-u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))
-H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
-(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2
-C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))) (ex2_2 C T
-(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0))))
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g
-a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_:
-(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
-T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
-T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
-c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
-(asucc g a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef
-i))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i)
-H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
-(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
-a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
-Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))
-H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1:
-(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2:
-A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5
-\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind
-T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
-(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind
-T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6
-(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
-Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2))))))
-(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
-(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
-(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
-u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda
-(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11:
-(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))
-x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C
-T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
-(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
-C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11:
-(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda
-(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
-T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
-(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2)
-(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
-
-theorem arity_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c:
-C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind
-b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_:
-A).(arity g (CHead c (Bind b) u) t a2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda
-(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity
-g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
-T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u
-a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y:
-T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda
-(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda
-(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
-a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n)
-(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind b) u t) H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u
-a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3)))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity
-g d u0 a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda
-(a1: A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t
-a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind
-(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
-(Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
-u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_:
-(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u
-a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g
-a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind
-(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
-(Bind b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0
-Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3:
-(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g
-(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u
-t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3))
-(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t
-a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u
-t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
-(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead
-k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead
-(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b0) u0 t0)
-(THead (Bind b) u t) H7) in ((let H10 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b0)
-u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda (H12:
-(eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead
-(Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u
-a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t
-a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g
-(CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def (eq_ind T u0
-(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: A).(arity g (CHead
-(CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) in (let H16 \def
-(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) t1) t a0)) H14 u
-H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b)
-u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
-(CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let H18 \def (eq_ind T u0
-(\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let H19 \def (eq_ind B b0
-(\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: A).(arity g (CHead
-(CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in (let H20 \def
-(eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t a0)) H16 b
-H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H2
-b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_:
-A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) H9))
-H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
-(H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 (THead (Bind b) u
-t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
-(CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: T).(\lambda (a0:
-A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (H5:
-(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead
-c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind
-Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead (Bind Abst) u0
-t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e
-in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _)
-\Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])]))
-(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
-(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
-(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0
-u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1:
-T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g
-(CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0
-(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0
-(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let
-H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to
-(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda
-(_: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u
-H10) in (let H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind
-Abst) t1) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1:
-T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u
-a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u
-H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g
-a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b0: B).((eq T t
-(THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind
-Abst) u) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u)
-(Bind b0) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda
-(b0: B).((eq T u (THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g
-c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1))))))
-H16 Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
-Abst))) H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t
-(AHead a1 a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) in False
-return (\lambda (_: False).(ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with
-[]) in H21) b H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda
-(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq
-T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0:
-T).(\lambda (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_:
-(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u
-a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1
-a0))))))).(\lambda (H6: (eq T (THead (Flat Appl) u0 t0) (THead (Bind b) u
-t))).(let H7 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)))
-H7)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_:
-(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t))
-\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g
-(CHead c0 (Bind b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_:
-(arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b)
-u t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
-A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b)
-u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
-(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1
-a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T
-(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0
-(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
-a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda
-(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7
-(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g
-c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11:
-(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g
-c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10
-(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y
-a2 H1))) H0)))))))).
-
-theorem arity_gen_abst:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
-A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead c (Bind Abst) u) t a2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
-A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead
-(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A
-A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1:
-A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2:
-A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0:
-(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
-A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n:
-nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H1) in
-(False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A (ASort O n)
-(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
-a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
-a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0:
-A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst)
-u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1
-a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda
-(_: A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
-(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
-i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g
-a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A
-(\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
-(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_:
-A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
-(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
-i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1:
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
-A).(\lambda (H2: (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind
-Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead
-a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
-(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
-a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0
-(Bind b) u0) t0 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b)
-u0) (Bind Abst) u) t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0)
-(THead (Bind Abst) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e
-in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
-(Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
-(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
-(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0
-u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1:
-T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t
-H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind
-b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T
-t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
-A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead
-c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let
-H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2))
-H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead
-(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
-a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0
-t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t
-(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
-A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0
-(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let
-H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2))
-H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
-Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in
-False return (\lambda (_: False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4:
-A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
-(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
-Abst) u) t a4))))) with []) in H21))))))))))))) H8)) H7))))))))))))))
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0
-u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2
-a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
-(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
-a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0
-(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc
-g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
-Abst) u0) (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst)
-u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
-(TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind
-Abst) u0 t0) (THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
-(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T
-u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
-Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead
-a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0)
-u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0
-(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T
-t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in
-(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t))
-\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc
-g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
-Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0
-(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let
-H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3
-a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))
-H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc
-g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A
-(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
-(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
-Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12)))))))))
-H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
-(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to
-(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda
-(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T
-(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T
-(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
-A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
-a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0:
-A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
-(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A
-(asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u
-(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind
-Abst) u) t a2))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
-a0)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
-(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
-(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity
-g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Cast)
-u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t)
-H5) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0
-(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
-a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
-a2)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
-A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Bind
-Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead
-a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2))))
-(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t
-a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T
-t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0
-(THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t1:
-T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u t) H5) in (let H7
-\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Bind Abst)
-u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind Abst) u t))) in
-(ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A
-(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0: A).(\lambda
-(x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10: (arity g c0 u
-(asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u) t x1)).(let
-H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in
-(let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u
-t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12)
-in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq
-g x0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
-(CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda
-(H15: (leq g x0 x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2
-(AHead x2 x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2
-x3) H17) in (eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda
-(a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda
-(_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity
-g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3:
-A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
-(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead
-x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2
-H15)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18)))))))
-H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
-
-theorem arity_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2:
-A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity
-g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2:
-A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead
-(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A
-(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1
-a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda
-(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t))
-\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
-(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
-(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1:
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O
-n))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a:
-A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl)
-u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g
-d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u
-t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u
-t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
-A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
-(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g
-d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda
-(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef
-i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1:
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a))))
-H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
-a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
-(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
-a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
-(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
-(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) u a3)) (\lambda (a3:
-A).(arity g (CHead c0 (Bind b) u0) t (AHead a3 a0))))))).(\lambda (H6: (eq T
-(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T
-(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Appl) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda
-(c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc
-g a1))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
-(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (asucc g
-a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
-(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
-(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda
-(a3: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a3 a0))))))).(\lambda
-(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def
-(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1
-a0))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
-A).(\lambda (H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat
-Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
-A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0:
-A).(\lambda (H3: (arity g c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0
-(THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))))).(\lambda (H5:
-(eq T (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t))).(let H6 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
-\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in
-((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _
-t1) \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5)
-in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq
-T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let
-H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t
-H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat
-Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
-A).(arity g c0 t (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0
-(\lambda (t1: T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3:
-A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12
-H10))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a:
-A).(\lambda (_: (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead
-(Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda
-(a1: A).(arity g c0 t (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda
-(_: (arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
-(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
-(AHead a1 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat
-Appl) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
-\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
-False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H5) in
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
-g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0:
-T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T
-t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
-(\lambda (a3: A).(arity g c0 t (AHead a3 a1))))))).(\lambda (a0: A).(\lambda
-(H3: (leq g a1 a0)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5
-\def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let
-H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to
-(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t
-(AHead a3 a1)))))) H2 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T
-t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in
-(let H8 \def (H6 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
-a1))) (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0
-t (AHead a3 a0)))) (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda
-(H10: (arity g c0 t (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0
-u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t
-(AHead x a1) H10 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3))))))
-H8))))))))))))) c y a2 H0))) H)))))).
-
-theorem arity_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
-A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a))
-(arity g c t a)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
-A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead
-(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land
-(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0:
-(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
-A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0))
-(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
-(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u
-(asucc g (ASort O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
-a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u
-(asucc g a0)) (arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat
-Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
-t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
-H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0:
-A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
-(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t
-(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u
-t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
-t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
-H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
-a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u
-(asucc g a1)) (arity g c0 t a1))))).(\lambda (t0: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T
-t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind b) u0) u
-(asucc g a2)) (arity g (CHead c0 (Bind b) u0) t a2))))).(\lambda (H6: (eq T
-(THead (Bind b) u0 t0) (THead (Flat Cast) u t))).(let H7 \def (eq_ind T
-(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g c0 t
-a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead
-(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a1))) (arity g c0
-t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g
-(CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast)
-u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g
-(CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0
-t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t)
-H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t
-(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda
-(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
-Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead
-a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g
-c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5:
-(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def
-(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
-\Rightarrow False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land
-(arity g c0 u (asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0:
-C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g
-a0))).(\lambda (H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0
-u (asucc g (asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0:
-T).(\lambda (H3: (arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat
-Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t
-a0))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Cast) u
-t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
-_ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t)
-H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
-| (THead _ _ t1) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat
-Cast) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0
-(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
-(asucc g a0)) (arity g c0 t a0)))) H4 t H7) in (let H10 \def (eq_ind T t0
-(\lambda (t1: T).(arity g c0 t1 a0)) H3 t H7) in (let H11 \def (eq_ind T u0
-(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
-(asucc g (asucc g a0))) (arity g c0 t (asucc g a0))))) H2 u H8) in (let H12
-\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a0))) H1 u H8) in
-(conj (arity g c0 u (asucc g a0)) (arity g c0 t a0) H12 H10)))))))
-H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
-(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u t))
-\to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (a2:
-A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u
-t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Cast) u t)
-H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat
-Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2
-(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
-T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6
-(refl_equal T (THead (Flat Cast) u t))) in (land_ind (arity g c0 u (asucc g
-a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2))
-(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t
-a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u
-(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1
-H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))).
-
-theorem arity_gen_appls:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall
-(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a:
-A).(arity g c t a))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs:
-TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads
-(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda
-(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c
-t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall
-(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a:
-A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead
-(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g
-c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g
-c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1
-a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_:
-(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x
-a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A
-(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a)))
-(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a:
-A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
-
-theorem arity_gen_lift:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h:
-nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2:
-C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T
-(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall
-(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda
-(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0))
-\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat
-d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2:
-C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c:
-C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq
-T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
-a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0:
-T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda
-(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0
-(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1)))))))))
-(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2:
-(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u
-(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
-a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i)
-(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
-(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
-T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
-(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
-i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
-(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
-(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
-nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
-in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
-i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
-(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
-(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
-(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
-\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
-t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
-a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u
-(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in
-(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h
-(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h
-(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i)
-(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef
-(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda
-(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda
-(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h)
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0
-H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst)
-u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda
-(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
-(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda
-(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x
-x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
-(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
-T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
-(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
-i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
-(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
-(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
-nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
-in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
-i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
-(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
-(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
-(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
-\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
-t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
-(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def
-(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus
-x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1
-(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt
-Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7:
-(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x
-h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le
-(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T
-(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0
-u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5
-H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1:
-(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall
-(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
-(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4:
-(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x:
-nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h
-x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x:
-nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x
-x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda
-(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0:
-T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda
-(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u
-(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T
-(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def
-(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
-(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to
-(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T
-t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x)
-x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c
-(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def
-(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift
-h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind
-b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15
-\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T
-t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
-a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1:
-T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1
-(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal
-T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b
-x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6))))))))))))))))))
-(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u
-(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u
-(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
-(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g
-(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall
-(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c
-(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda
-(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda
-(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0:
-T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0:
-T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1:
-T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1
-x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S
-x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2
-t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3:
-nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h
-x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x)
-x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c
-(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u
-(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11
-(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3:
-nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall
-(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2)))))))
-H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall
-(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3:
-C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1)
-H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g
-a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T
-(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2))
-(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0
-H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda
-(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
-(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4:
-((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall
-(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda
-(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift
-h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T
-(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
-(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1:
-T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1
-x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x
-x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1
-a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
-(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def
-(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2)
-H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
-(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
-(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2
-x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2
-(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0
-x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
-A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x:
-nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x
-c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3:
-(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T
-t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
-a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead
-(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c
-c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
-Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0))))
-(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0)
-(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast)
-x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h
-x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1
-a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
-(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0
-(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def
-(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
-(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
-(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
-(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in
-(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10
-x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast
-u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
-A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall
-(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
-(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
-a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x
-x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1
-(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/arity/props.ma".
-
-include "LambdaDelta-1/drop1/fwd.ma".
-
-theorem arity_lift1:
- \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds:
-PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t
-a) \to (arity g c1 (lift1 hds t) a))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds:
-PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t:
-T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))
-(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda
-(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C
-c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1:
-C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1
-(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1
-(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def
-(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
-(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1
-(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1
-x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1)
-c1 n n0 H3)))) H2))))))))))) hds)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csuba/arity.ma".
-
-include "LambdaDelta-1/pr3/defs.ma".
-
-include "LambdaDelta-1/pr1/defs.ma".
-
-include "LambdaDelta-1/wcpr0/getl.ma".
-
-include "LambdaDelta-1/pr0/fwd.ma".
-
-include "LambdaDelta-1/arity/subst0.ma".
-
-theorem arity_sred_wcpr0_pr0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
-c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1
-t2) \to (arity g c2 t2 a)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
-(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to
-(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2:
-C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n)
-t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n)))
-(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda
-(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d
-(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda
-(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to
-(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c
-c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i)
-(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda
-(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2
-(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2
-(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u
-x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2
-H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g
-a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2:
-T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2:
-C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i)
-t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
-T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0
-d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1
-H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i
-H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda
-(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u
-a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
-u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda
-(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2:
-C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to
-(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c
-c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq
-T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g
-c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0:
-T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2))))
-(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def
-(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T
-(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0
-c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp
-c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1
-(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda
-(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b)
-u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k
-u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
-(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
-(THead (Bind b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3)
-(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
-(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3)
-(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K
-k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2
-t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind
-b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3
-(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1
-(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9
-u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16)
-in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b)
-u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14))
-H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda
-(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g
-c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
-t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4
-a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
-(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead
-(Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2)
-H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
-(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda
-(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u
-t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_:
-(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4
-a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3))
-(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead
-(Bind b0) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda
-(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u
-t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity
-g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda
-(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3)
-(THead (Bind b) u t) H13) in ((let H15 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind
-Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0]))
-(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T
-u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0:
-T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in
-(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let
-H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to
-(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0:
-T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0:
-B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in
-(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t))
-\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b
-(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to
-(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let
-H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2))
-H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
-Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w
-a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr)
-u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr
-c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda
-(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9:
-(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2
-t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S
-O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 |
-(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in
-((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0
-_) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u
-t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
-\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t5)
-\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in
-lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
-t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0)
-\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t)
-H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def
-(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18
-\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to
-(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T
-t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to
-(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O
-t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c
-(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2
-(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u
-(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2
-(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12))))))))))
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_:
-(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0:
-T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let
-H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2)
-H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
-((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g
-c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2:
-(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2:
-C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to
-(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
-c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t)
-t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2))
-(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6:
-(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind
-Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda
-(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda
-(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u
-t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1
-c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u
-u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1
-(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
-(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k:
-K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _)
-\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0)
-\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda
-(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind
-Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17
-\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to
-(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3
-(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1
-(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead
-a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0
-u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead
-c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k
-H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t))
-\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t))
-\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl)
-v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind
-T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead
-(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda
-(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
-v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2
-(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
-u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2
-(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
-t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4
-(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b)
-u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl)
-v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda
-(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1
-(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead
-(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w:
-T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr)
-u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T (THead (Bind Abbr)
-u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g
-c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b:
-B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u
-t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq
-T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11
-\def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B)
-with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O
-t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
-(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b)
-u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T
-\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
-t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T
-\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
-t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0)
-\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u
-t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let
-H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0))
-\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18
-\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind
-Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3
-(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity
-g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def
-(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g
-c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda
-(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
-(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0:
-T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u
-t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t) H9) in (False_ind (arity g c2
-t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda
-(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to
-(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2:
-(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2)
-\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1
-a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2:
-T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat
-Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2))
-(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda
-(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda
-(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def
-(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T
-(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2
-u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8))))
-(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8:
-(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
-(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda
-(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal
-T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
-(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0]))
-(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0]))
-(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1
-u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda
-(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda
-(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t
-H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in
-(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u
-t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda
-(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4
-a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0:
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8:
-(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3
-(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T
-(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u
-t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead
-_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
-(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead
-(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead
-_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
-(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def
-(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g
-c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0
-v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3
-(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0
-t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead
-(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3)
-H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0
-c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1
-a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t
-(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3)
-H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead
-(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in
-(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A
-(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4))))
-(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_:
-A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2
-(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda
-(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0
-(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let
-H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
-with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1
-a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match
-e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _
-a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq
-A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2
-(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda
-(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr
-not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4
-a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1
-H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b:
-B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u
-t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda
-(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to
-(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0
-t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4
-a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
-\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead
-(Bind b) u1 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T
-v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat
-Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1
-(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t
-(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2)))
-H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0:
-T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead
-(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u
-(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3)
-H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0
-c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1
-a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t
-(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16)
-in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind
-b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def
-(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda
-(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2)
-t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda
-(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7
-c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead
-c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2
-(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2
-H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2:
-T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t))
-\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
-t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4
-a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T
-(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T
-(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Appl) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2)
-H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
-(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda
-(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u
-t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind
-(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda
-(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity
-g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3)
-(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t)
-H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5))))))))))))))
-(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u
-(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall
-(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t:
-T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c
-c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda
-(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0
-(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda
-(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y:
-T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq
-T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0:
-T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T
-(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat
-Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2
-H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1
-(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda
-(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat
-Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T
-(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda
-(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
-| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
-(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3)
-(THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
-(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3)
-(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16:
-(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2
-(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0
-(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def
-(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind
-T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2
-a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2))
-H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4
-H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead
-(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t))
-\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead
-(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead
-(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
-\Rightarrow False])])])) I (THead (Flat Cast) u t) H11) in (False_ind (arity
-g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\lambda (b: B).(\lambda
-(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
-v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2
-a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
-(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let
-H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t)
-H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2:
-T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t))
-\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
-t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4
-a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T
-(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \def (eq_ind T
-(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Cast) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0)
-H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3
-(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda
-(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u
-t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H10) in (False_ind
-(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to
-(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast)
-u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
-(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat
-Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0]))
-(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T
-u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat
-Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3
-(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10))))))))
-y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1:
-A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c
-c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda
-(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c
-c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2
-H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))).
-
-theorem arity_sred_wcpr0_pr1:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall
-(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1
-c2) \to (arity g c2 t2 a)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a:
-A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0
-a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a:
-A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1
-c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda
-(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a:
-A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5
-a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3:
-(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a
-(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl
-c2)))))))))))))) t1 t2 H))).
-
-theorem arity_sred_pr2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g:
-G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda
-(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda
-(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0
-g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
-G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a
-(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t
-H2)))))))))))))) c t1 t2 H)))).
-
-theorem arity_sred_pr3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
-(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a:
-A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g:
-G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
-(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c
-t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3:
-(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2
-H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/arity/fwd.ma".
-
-theorem node_inh:
- \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c:
-C).(\lambda (t: T).(arity g c t (ASort k n)))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0:
-nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n))))))
-(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n))))
-(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda
-(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0
-n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t:
-T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t:
-T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c:
-C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1)
-(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0
-x1) (ASort (S n0) n) H1))))) H0)))) k))).
-
-theorem arity_lift:
- \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2
-t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1
-c2) \to (arity g c1 (lift h d t) a)))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
-A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to
-(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n:
-nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop
-h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O
-n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d))))))))
-(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1:
-(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall
-(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda
-(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1
-c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i
-d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def
-(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d
-(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
-O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
-(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity
-g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O
-c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1
-(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n:
-nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let
-H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C
-(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i))
-u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i)
-a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h
-(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x
-d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead
-x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S
-i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
-H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0:
-T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i
-c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i))
-(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
-Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g
-a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0:
-nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g
-a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
-(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0)
-(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1
-t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0
-H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0:
-C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop
-h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d
-(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1:
-C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0
-x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind
-nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
-(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
-H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind
-Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S
-i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0
-(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h
-(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i
-(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11)
-a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i))
-(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus
-i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h)
-(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1)
-(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
-(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
-(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
-g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1
-(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u)
-(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind
-g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4
-(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5
-b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h
-d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
-(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g
-a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
-(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1
-(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d
-u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1
-a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst)
-d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d)
-(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0))
-(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
-(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
-(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
-g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead
-a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s
-(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1
-(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s
-(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat
-Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
-A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1:
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift
-h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
-a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1:
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1
-c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d)
-t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0
-(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4))
-(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h
-d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
-(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0)
-a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1:
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1
-c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a
-H))))).
-
-theorem arity_mono:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c
-t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H:
-(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
-A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0:
-C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n)
-a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u
-a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a
-a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
-\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
-(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
-Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
-(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
-(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
-x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind
-Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _)
-\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1)
-(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in
-((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead
-d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr)
-u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d x0)).(let
-H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abbr)
-t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0
-t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
-(CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
-(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9)))))))
-H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
-(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
-(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
-(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g
-a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i
-c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i
-H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind
-Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind
-Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst)
-x1) H6)) in (False_ind (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c0 (CHead d (Bind Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u
-(asucc g a))).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g
-(asucc g a) a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i)
-a2)).(let H4 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda
-(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
-(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2)
-(\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead
-d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
-a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
-(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
-(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
-x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind
-Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr)
-x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6))
-in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda
-(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
-(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T
-(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
-(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2)
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind
-Abst) x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind
-C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind
-Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst)
-x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
-c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead
-d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
-Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0
-(CHead x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def
-(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u
-H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g
-a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
-(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
-(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2
-(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
-T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall
-(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda
-(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4:
-((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3
-a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0)
-a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A
-(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind
-b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u
-x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8))))
-H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda
-(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0
-u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3:
-A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3:
-((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3
-a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0)
-a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda
-(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda
-(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity
-g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7:
-(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u)
-t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a))
-(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8))
-a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2:
-A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g
-c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_:
-(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0
-t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity
-g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0
-H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity
-g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g
-c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x
-a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u:
-T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_:
-((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda
-(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2:
-A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4:
-(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u
-t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g
-a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
-a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
-(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3:
-A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2:
-(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans
-g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
-
-theorem arity_repellent:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1:
-A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c
-(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P:
-Prop).P)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1:
-A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2:
-A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g
-a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst)
-w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind
-A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
-A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4:
-A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g
-x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def
-(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y
-(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind
-Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))).
-
-theorem arity_appls_cast:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs:
-TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to
-((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl)
-vs (THead (Flat Cast) u t)) a))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs:
-TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads
-(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a)
-\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a)))))
-(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity
-g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1:
-TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u)
-(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c
-(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a:
-A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u))
-(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat
-Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1
-t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1:
-A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead
-(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda
-(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat
-Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat
-Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1))
-(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g
-a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat
-Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda
-(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g
-a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))
-a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g
-a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7
-(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc
-g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g
-(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))).
-
-theorem arity_appls_abbr:
- \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall
-(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c
-(THeads (Flat Appl) vs (TLRef i)) a)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs:
-TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads
-(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t
-(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v)
-a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop
-Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0:
-((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to
-(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda
-(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
-v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S
-i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1:
-A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity
-g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x:
-A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat
-Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat
-Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))).
-
-theorem arity_appls_bind:
- \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c:
-C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t:
-T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v)
-(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat
-Appl) vs (THead (Bind b) v t)) a2)))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
-(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v
-a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
-TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl)
-(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind
-b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v)
-t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1:
-TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads
-(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1
-(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead
-c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts
-(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v)
-(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in
-(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0)
-a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl)
-(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0
-(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda
-(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity
-g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x
-a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O)
-O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1
-(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/arity/props.ma".
-
-include "LambdaDelta-1/fsubst0/fwd.ma".
-
-include "LambdaDelta-1/csubst0/getl.ma".
-
-include "LambdaDelta-1/subst0/dec.ma".
-
-include "LambdaDelta-1/subst0/fwd.ma".
-
-include "LambdaDelta-1/getl/getl.ma".
-
-theorem arity_gen_cvoid_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
-(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to
-(\forall (P: Prop).P))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
-A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d
-(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
-(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda
-(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d
-(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w
-(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P)))))))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_:
-(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall
-(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall
-(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0:
-C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0
-(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w
-(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i)
-O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O
-w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0
-(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
-(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0
-(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d
-(Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9))))))
-(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda
-(_: ((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead
-d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v)
-\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda
-(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w:
-T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P:
-Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq
-nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat
-i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let
-H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0
-(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead
-d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Void)
-u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7))
-in (False_ind P H9)))))) (subst0_gen_lref w v i0 i H4))))))))))))))))))
-(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2:
-((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d
-(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to
-(\forall (P: Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_:
-(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d:
-C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d
-(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
-(\forall (P: Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
-nat).(\lambda (H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w:
-T).(\lambda (v: T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0)
-v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind
-b) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq
-T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0
-t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T
-(\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i
-w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0)))
-(\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v
-(THead (Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9
-P)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u
-t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda
-(t2: T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b)
-i) w t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u
-x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i)
-(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d
-(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u
-x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9
-P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6)))))))))))))))))))))
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
-(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
-nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
-T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
-a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl
-i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w:
-T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
-T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P:
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0)))
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
-(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6:
-(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2:
-T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind
-Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
-(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
-u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
-(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2)))
-(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x:
-T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s
-(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind
-Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8
-P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda
-(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda
-(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1
-d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i
-H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
-(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0
-t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall
-(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall
-(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind
-Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead
-(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq
-T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T
-(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0
-(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
-T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w
-u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0
-t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2
-t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T
-v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda
-(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0
-i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2:
-T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl)
-i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2)))
-(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x:
-T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s
-(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
-(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w
-u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
-P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5)))))))))))))))))))
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u
-(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
-nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
-T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
-(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
-T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P:
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0)))
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
-(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6:
-(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2:
-T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat
-Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
-(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
-u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
-(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2)))
-(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x:
-T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s
-(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
-(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w
-u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
-P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5))))))))))))))))))
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
-t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i:
-nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v:
-T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2:
-A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w:
-T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d
-u i H3 w v H4 P)))))))))))))))) c t a H))))).
-
-theorem arity_gen_cvoid:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
-(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in
-(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i
-v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i
-v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T
-t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift
-(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3:
-(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u
-(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))
-(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda
-(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i
-x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v)))))
-(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x
-(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
-
-theorem arity_fsubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
-c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u
-c1 t1 c2 t2) \to (arity g c2 t2 a))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
-(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
-d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2
-t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n:
-nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i
-c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1:
-(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort
-n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u
-(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i
-u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3:
-(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i
-u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c
-c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0:
-C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2
-(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0
-i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2
-(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c
-c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n)))
-(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n)
-t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c
-c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n)
-t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g
-c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
-Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2:
-((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2
-t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0:
-T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr)
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef
-i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in
-(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))
-(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i)
-t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2)
-(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i)
-t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0
-(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq
-nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat
-i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O
-u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
-d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind
-Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)
-(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in
-((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d
-(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
-i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d d1)).(let H16
-\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H12
-u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let
-H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u)))
-H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i
-H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7)))
-H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind
-(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq
-T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i)
-(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0)
-(\lambda (H9: (lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8
-(CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abbr) u))
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
-i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w:
-T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i))
-u0 e1 e2))))))) (arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d
-(Bind Abbr) u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n:
-nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)))
-(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0
-(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
-in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda
-(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind
-x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def
-(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u)
-(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3
-(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0
-(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in
-(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def
-(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18)
-in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind
-x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i
-c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0
-(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead
-d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr)
-(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda
-(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b)
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda
-(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq
-C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2
-(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1
-x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
-d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
-Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
-(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u)
-(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
-b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
-(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let
-H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t)))
-H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus
-i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u
-i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind
-Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11))
-(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
-x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i)
-(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)))
-(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0
-(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
-in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0]))
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H19
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr
-x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
-T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
-x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
-H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
-H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr)
-(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u
-(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0
-d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
-(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
-d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
-(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
-(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
-(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
-T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
-nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
-d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind
-Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)
-(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in
-((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d
-(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
-i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17
-\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13
-u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2))
-H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0))
-(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr)
-u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u
-i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0
-H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6))
-H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
-A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1:
-C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0))
-\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g
-c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0:
-nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2:
-C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x
-\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in
-(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i)
-t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c
-c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef
-i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0)
-(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind
-C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2
-(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
-(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
-T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n
-c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d
-(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0)
-(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in
-(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
-C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d
-(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c
-(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2
-H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c
-c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0)
-(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c
-c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0
-(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def
-(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in (or4_ind
-(getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0)
-(\lambda (H11: (getl i c2 (CHead d (Bind Abst) u))).(let H12 \def (eq_ind nat
-(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1
-(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d
-(Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i)))
-(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11:
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda
-(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
-Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0:
-B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2
-(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2
-x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
-d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
-(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u)
-(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
-b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u)
-(CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let
-H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3))
-H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead
-c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3
-i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind
-Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11))
-(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
-(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b)
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda
-(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq
-C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2
-(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1
-x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
-d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
-(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u)
-(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
-b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u)
-(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let
-H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t)))
-H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus
-i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u
-i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind
-Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11))
-(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
-x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i)
-(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0)))
-(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0
-(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
-in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0]))
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H19
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst
-x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t:
-T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C
-x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let
-H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4)))
-H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst)
-(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u
-(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0
-d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
-(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
-d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
-(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
-(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
-(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
-T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
-nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead
-d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind
-Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr)
-u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12))
-in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10)))
-(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall
-(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr)
-u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to
-(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
-(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall
-(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr)
-u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b)
-u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0:
-T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr)
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead
-(Bind b) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u
-t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0
-(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0
-c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2))
-(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind
-b) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2)
-(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0
-(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2))
-(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda
-(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b)
-u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T
-(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
-u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t)))
-(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x:
-T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0
-u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2))
-(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t
-a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b
-c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S
-i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u
-x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2
-(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
-(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x:
-T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s
-(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity
-g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i)
-(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
-(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c
-(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda
-(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t
-x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2))
-(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13))
-x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind
-b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both
-(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0)
-(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head
-(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead
-(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t)
-t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind
-b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u
-t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0
-i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i)
-(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
-(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c
-(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u))))
-t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2)
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2)
-(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead
-(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T
-(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
-u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
-(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2
-T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0
-i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t)))
-(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x:
-T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0
-u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2))
-(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13
-c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u
-(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t
-(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x)
-(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda
-(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3:
-T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2
-(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))
-(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u
-x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind
-b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2
-d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i)
-(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
-(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c
-(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10
-u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b)
-i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)))
-(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2
-(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14:
-(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda
-(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0
-(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind
-b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5)
-(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1
-H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2
-H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8))
-H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1:
-C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0))
-\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g
-c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
-(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1:
-C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead
-d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0
-(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda
-(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1
-(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i
-u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2
-(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq
-C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind
-Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u
-t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land
-(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2)
-(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2))
-(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t)
-t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind
-(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2:
-T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u
-t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2
-(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind
-Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
-T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))
-(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead
-(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind
-Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x
-a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i)
-(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i)
-u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i
-u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq
-T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0
-t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3)))
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1
-a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u
-x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead
-(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g
-c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u)
-c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind
-Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2
-H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
-u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity
-g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T
-t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda
-(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0
-x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1
-d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i)
-(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S
-i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0)
-(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10))
-(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7:
-(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T
-(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2))
-(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0
-c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0
-(AHead a1 a2))) (arity_head g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c
-u c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u)
-c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind
-Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind
-Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda
-(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c
-c2))).(land_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2)
-(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u
-t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2:
-T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))
-(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3:
-T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
-Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T
-(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0
-i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t)))
-(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 (AHead a1 a2)) (\lambda
-(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12:
-(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0:
-T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x
-(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind
-Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr)
-u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind
-Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2
-H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
-(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3)))
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1
-a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u
-x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead
-(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head
-g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S
-i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S
-i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u)
-(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2
-(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2
-(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
-(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
-(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1
-u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i)
-(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S
-i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0)
-(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10))
-(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6)))))))))))))))))))
-(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u
-a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
-nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t:
-T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3:
-((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
-t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0:
-T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
-(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat
-Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2)
-(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t)
-t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2)
-(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2)
-(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0
-(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c
-c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c
-(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T
-t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T
-(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0
-(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
-u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
-t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2
-(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0
-i u0 u u2)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead
-(Flat Appl) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat
-Appl) x t) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0
-i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda
-(H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda
-(t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq
-T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0
-t t3)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat
-Appl) u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T
-(THead (Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u
-a1 H0 x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10))
-(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity
-g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
-(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
-(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
-(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0
-(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c
-t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9))
-c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0
-i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)
-(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda
-(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0:
-T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst
-i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
-H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2)
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2)
-(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead
-(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
-(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0
-i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
-(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10:
-(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2:
-T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat
-Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2)
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda
-(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0:
-T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x
-(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i
-u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T
-t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
-(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2)
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda
-(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x)
-(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2
-u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c
-t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0
-x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
-Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0:
-T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0
-(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1
-(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head
-(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g
-a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
-nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
-a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3:
-((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2
-t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0:
-T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
-(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat
-Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2)
-(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t)
-t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2)
-(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2)
-(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0
-(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c
-c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c
-(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T
-t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T
-(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0
-(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
-u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
-t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2
-(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0
-i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead
-(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat
-Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0
-i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10:
-(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3:
-T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2
-(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
-t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat
-Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T
-(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u
-a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10))
-(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity
-g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
-(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
-(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
-(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0
-(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t
-x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2
-H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i
-u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)
-(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda
-(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0:
-T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst
-i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
-H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2)
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2)
-(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead
-(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
-(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0
-i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
-(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10:
-(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2:
-T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat
-Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0)
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda
-(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0:
-T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x
-(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0
-c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
-(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
-(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0)
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda
-(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x)
-(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2
-u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t
-x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0
-x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
-Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0:
-T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0
-(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i
-u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t
-t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t:
-T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall
-(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr)
-u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity
-g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda
-(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1
-(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u
-c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def
-H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2)
-(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2
-t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c
-c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda
-(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2))
-(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2)
-c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind
-(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t
-t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g
-c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2
-H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u
-c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2)
-(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c
-c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7
-c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))).
-
-theorem arity_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c
-t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
-d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2
-a)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H:
-(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
-(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u
-c t1 t2 H1)))))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/A/defs.ma".
-
-include "LambdaDelta-1/G/defs.ma".
-
-definition asucc:
- G \to (A \to A)
-\def
- let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n)
-\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h)
-\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g
-a2))]) in asucc.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/asucc/defs.ma".
-
-theorem asucc_gen_sort:
- \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A
-(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
-nat).(eq A a (ASort h0 n0)))))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind
-(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda
-(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0:
-nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0
-n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0
-with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0
-n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A
-(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1)))))))
-(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat
-nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0
-n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1))
-\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0
-n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2
-\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda
-(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda
-(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2)))))))
-a)))).
-
-theorem asucc_gen_head:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A
-(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1
-a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind
-(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3:
-A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3))))))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc
-g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g
-(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1
-a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead
-a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0))
-H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0)))
-(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda
-(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0:
-A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
-a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1)
-n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
-_) \Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0:
-A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
-a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2)
-(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda
-(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A
-(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1
-a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead
-a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
-(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in
-((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_:
-A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4]))
-(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let
-H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4))
-\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A
-a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A
-(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A
-a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead
-a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5)))
-(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7
-\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to
-(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4
-(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda
-(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda
-(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead
-a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3
-(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4))))
-H2))))))) a)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/defs.ma".
-
-definition cimp:
- C \to (C \to Prop)
-\def
- \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall
-(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C
-(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/cimp/defs.ma".
-
-include "LambdaDelta-1/getl/getl.ma".
-
-theorem cimp_flat_sx:
- \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v)
-c)))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
-C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f)
-v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat
-f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2
-(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind
-b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1
-(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c
-(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b)
-w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v)
-(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind
-b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind
-b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w)))
-d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))).
-
-theorem cimp_flat_dx:
- \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f)
-v))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
-C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind
-b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2
-(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
-
-theorem cimp_bind:
- \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
-(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
-C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
-(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
-(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w:
-T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1
-(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v)
-(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b)
-v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v)
-(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
-\Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind
-b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1
-(Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in
-C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead
-c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
-(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead
-c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
-(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0
-b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda
-(d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B
-b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v)
-(CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2
-(Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4))))
-H3)) H2))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c1 (Bind b) v)
-(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl h0 (CHead c2 (Bind
-b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind
-b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0)
-(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x
-in (ex_ind C (\lambda (d2: C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C
-(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))))
-(\lambda (x: C).(\lambda (H3: (getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C
-(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))
-x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w) H3 v)))) H2)))))) h
-H0)))))))))).
-
-theorem cimp_getl_conf:
- \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
-(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w))
-\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead
-d2 (Bind b) w)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
-C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
-(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
-(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def
-H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C
-(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall
-(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4:
-C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x
-(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3:
-C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0))
-\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0))))))))))
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0:
-B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h
-d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1
-(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0
-(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in
-(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2
-(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind
-b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0
-(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind
-b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat
-(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n:
-nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0
-(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def
-(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind
-b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro
-C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind
-b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-inductive clear: C \to (C \to Prop) \def
-| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e
-(Bind b) u) (CHead e (Bind b) u))))
-| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f:
-F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/clear/fwd.ma".
-
-include "LambdaDelta-1/drop/fwd.ma".
-
-theorem drop_clear:
- \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to
-(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead
-e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
-c2))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i:
-nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
-C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda
-(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda
-(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind
-(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b:
-B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v)))))
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda
-(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat
-O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e:
-C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_:
-B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort
-n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall
-(i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
-C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda
-(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k
-t) c2)).(K_ind (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T
-(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead
-e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
-c2))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) i) O c
-c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v:
-T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_:
-B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c
-t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2
-\def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda
-(v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e:
-C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda
-(e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v)))))
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda
-(x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1
-(Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda
-(b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e
-(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
-c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2))))
-k (drop_gen_drop k c c2 t i H0))))))))) c1).
-
-theorem drop_clear_O:
- \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c
-(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1
-e2) \to (drop (S i) O c e2))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1:
-C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
-C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2))))))))
-(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort
-n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_:
-(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O
-(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1:
-C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
-C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0
-e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u:
-T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2:
-C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0:
-K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0
-k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t)
-(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _)
-\Rightarrow c1])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t)
-(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal
-C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_:
-K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1
-(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b)
-u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow
-t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0
-(CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq
-C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: C).(drop i O c1 e2)) H1 c0
-H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2))
-(drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f:
-F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b)
-u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind
-b) u) t H2) e2 i H1) t))) k H0))))))))))) c)).
-
-theorem drop_clear_S:
- \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop
-h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear
-x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1
-(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))
-\def
- \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h:
-nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
-C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
-C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
-c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda
-(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b)
-u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1:
-C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
-c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h:
-nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
-C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
-C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
-c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k
-t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear
-(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1
-(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C
-(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1:
-C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift
-h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k
-(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead
-c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind
-(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h
-(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d)
-t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))
-(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind
-b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u)
-(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in
-((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
-C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in
-K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
-\Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t)
-(clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C
-T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead
-c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda
-(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0:
-T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d)
-t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2))))
-(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind
-b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda
-(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda
-(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind
-b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda
-(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind
-b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x
-(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda
-(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u
-(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1:
-C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
-c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d)
-t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))
-(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d
-u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear
-(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d
-u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b)
-(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1
-H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/clear/defs.ma".
-
-theorem clear_gen_sort:
- \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P:
-Prop).P)))
-\def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda
-(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_:
-C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c:
-C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let
-H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
-\Rightarrow True])) I (CSort n) H1) in (False_ind P H2)))))) (\lambda (e:
-C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (_: (((eq C e (CSort
-n)) \to P))).(\lambda (f: F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e
-(Flat f) u) (CSort n))).(let H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
-(False_ind P H4))))))))) y x H0))) H)))).
-
-theorem clear_gen_bind:
- \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
-(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
-\def
- \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
-(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda
-(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0:
-(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e
-(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda
-(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b)
-u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow
-c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H3 \def
-(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def
-(f_equal C T (\lambda (e1: C).(match e1 in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind
-b0) u0) (CHead e (Bind b) u) H1) in (\lambda (H5: (eq B b0 b)).(\lambda (H6:
-(eq C e0 e)).(eq_ind_r T u (\lambda (t: T).(eq C (CHead e0 (Bind b0) t)
-(CHead e0 (Bind b0) t))) (eq_ind_r C e (\lambda (c: C).(eq C (CHead c (Bind
-b0) u) (CHead c (Bind b0) u))) (eq_ind_r B b (\lambda (b1: B).(eq C (CHead e
-(Bind b1) u) (CHead e (Bind b1) u))) (refl_equal C (CHead e (Bind b) u)) b0
-H5) e0 H6) u0 H4)))) H3)) H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda
-(_: (clear e0 c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c
-e0)))).(\lambda (f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat
-f) u0) (CHead e (Bind b) u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (CHead e (Bind b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f)
-u0)) H4))))))))) y x H0))) H))))).
-
-theorem clear_gen_flat:
- \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
-(CHead e (Flat f) u) x) \to (clear e x)))))
-\def
- \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
-(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda
-(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda
-(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead
-e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f)
-u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
-_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1)
-in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda (e0:
-C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0
-(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0:
-T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4
-\def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0])) (CHead e0
-(Flat f0) u0) (CHead e (Flat f) u) H3) in ((let H5 \def (f_equal C F (\lambda
-(e1: C).(match e1 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow
-f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) with
-[(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0)
-u0) (CHead e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1:
-C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u)
-H3) in (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let H9 \def
-(eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to (clear e
-c))) H2 e H8) in (let H10 \def (eq_ind C e0 (\lambda (c0: C).(clear c0 c)) H1
-e H8) in H10))))) H5)) H4))))))))) y x H0))) H))))).
-
-theorem clear_gen_flat_r:
- \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x
-(CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
-\def
- \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H:
-(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e
-(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y:
-C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0:
-C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat
-f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match
-ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat
-f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda
-(H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to
-P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f)
-u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f)
-u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda
-(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C
-(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
-
-theorem clear_gen_all:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind
-(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0:
-B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0
-(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e:
-C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b)
-u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b)
-u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c
-(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c
-(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C
-(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B
-C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind
-x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0)
-x2))) c H3)))))) H2)))))))) c1 c2 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/clear/fwd.ma".
-
-theorem clear_clear:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
-(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear
-(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c:
-C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2
-c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear
-(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to
-(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t)
-c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0))
-(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f:
-F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c
-c2 t H1)))) k H0))))))) c1).
-
-theorem clear_mono:
- \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c
-c2) \to (eq C c1 c2)))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to
-(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n:
-nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2:
-C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1
-c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to
-(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t)
-c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind
-(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2)
-\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b)
-t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0
-(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t)
-(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0
-(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t
-H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t)
-c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f
-c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
-
-theorem clear_trans:
- \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c
-c2) \to (clear c1 c2)))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to
-(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n:
-nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2:
-C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n)
-c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to
-(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t)
-c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0:
-K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b:
-B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0
-(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c
-c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead
-c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3)))))
-(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c
-c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
-
-theorem clear_ctail:
- \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1
-(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k
-u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
-\def
- \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
-C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
-K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b)
-u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H:
-(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1:
-T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1
-c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n
-H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind
-b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear
-(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2))))
-k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2:
-T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1:
-T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear
-(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1:
-T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to
-(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2))))
-(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind
-b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t)
-(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal
-C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_:
-K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2
-(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b)
-u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0)
-\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t)
-(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b
-b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead
-(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r
-C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail
-k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0
-u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail
-k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1:
-(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1
-c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead
-c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)).
-
-theorem clear_cle:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
-(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda
-(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O)))))
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight
-c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
-C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear
-(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t)))))
-(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C
-(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c)
-(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c
-c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t)
-c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2
-(clear_gen_flat f c c2 t H1))))) k H0))))))) c1).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-include "LambdaDelta-1/s/defs.ma".
-
-definition clen:
- C \to nat
-\def
- let rec clen (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O |
-(CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/clen/defs.ma".
-
-include "LambdaDelta-1/getl/props.ma".
-
-theorem getl_ctail_clen:
- \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n:
-nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
-\def
- \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex
-nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n)
-(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O
-(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b
-(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl
-(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k:
-K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl
-(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat
-(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0)
-(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen
-c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0:
-K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t
-c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat
-(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0)
-t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail
-(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f:
-F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t
-c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b)
-t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))).
-
-theorem getl_gen_tail:
- \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall
-(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2
-(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
-(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
-nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
-nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
-\def
- \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda
-(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i
-(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
-c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4
-nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
-n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0:
-nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C
-(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n)
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort
-n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2))
-(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead
-(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear
-(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e:
-C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_:
-nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
-nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead
-(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead
-(CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2)
-u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow
-(match k0 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
-(Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0)
-u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2
-(Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n)
-(CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5:
-(eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C
-(\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O
-(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O))
-(\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2))
-(\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or
-(ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e:
-C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq
-nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq
-T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0
-(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0)
-u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat
-(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1)))
-(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort
-n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0)
-u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat
-(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0)))
-(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort
-n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K
-(Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq
-C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0))
-(refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2))
-H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead (CSort n) (Flat f) u1)
-(CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n
-(clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda
-(e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n)
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda
-(_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
-(n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) k u1)
-(CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 (CHead
-(CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
-c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b)
-u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: nat).(eq K k
-(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort
-n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind
-b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k
-(CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda (e: C).(eq C
-c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b)
-u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K
-k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2
-(CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: ((\forall (i:
-nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda
-(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b)
-u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq
-K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2
-(CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(nat_ind
-(\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b)
-u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
-C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
-nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b)))
-(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))
-(\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b)
-u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2
-(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
-(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat
-(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
-n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind
-b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
-(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c)
-(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1))
-in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
-(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow (match
-k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
-\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t)
-(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2
-(Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k
-u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda
-(H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C
-(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c
-(Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O
-(s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
-nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b
-(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
-(\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4
-nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_:
-nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
-C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i0:
-nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda
-(e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i0 c (CHead e (Bind b)
-u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 (clen c))) (\lambda (_: nat).(eq
-K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0
-(CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda
-(c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e:
-C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda
-(_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort
-n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1
-e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2))))
-(ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_:
-nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
-C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1
-c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e
-(Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6))
-b0 H5) t H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead
-(CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O
-(getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl
-(CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t
-H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda
-(e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat
-O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
-u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C
-c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
-(n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2
-(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind
-C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead
-e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
-(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4
-nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
-K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2
-(CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1
-x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1
-x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e)))
-(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4
-nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
-K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0
-(CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail
-k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b)
-u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda
-(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n:
-nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C
-(CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t)
-(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x
-(Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_:
-nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
-nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat
-(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
-(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or
-(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O
-(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
-nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
-(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0:
-nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind
-b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort
-x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq
-C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
-(n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C
-(\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O
-(CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq
-nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
-(_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))))))
-(eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort
-x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
-(Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
-(\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1))
-(\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda
-(e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O
-(CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq
-nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
-(\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort
-n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
-(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1))
-(\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b))
-(refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3))
-H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2)
-H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead (CTail k u1 c) k0 t)
-(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
-e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
-(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort
-n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2
-(Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead
-c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (e:
-C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind
-b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_:
-nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
-nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
-e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4
-nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K
-k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2
-(CSort n0))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
-(\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C
-(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c
-(CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
-(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
-(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k
-(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort
-n0))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5:
-(getl (r k0 n) c (CHead x (Bind b) u2))).(let H6 \def (eq_ind C c2 (\lambda
-(c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0
-(CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) (CTail k u1 x) H4) in (let H7
-\def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t)
-(CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1
-e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
-(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort
-n0))))))) H0 (CTail k u1 x) H4) in (eq_ind_r C (CTail k u1 x) (\lambda (c0:
-C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl
-(S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
-nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
-nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl
-(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e:
-C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
-nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
-(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x)
-(CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1
-e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x
-(refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H5 t)))
-c2 H4)))))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n)
-(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
-u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))).(ex4_ind nat (\lambda (_:
-nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
-(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or
-(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n)
-(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S
-n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
-nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0:
-nat).(\lambda (H4: (eq nat (r k0 n) (clen c))).(\lambda (H5: (eq K k (Bind
-b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(let H8
-\def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0
-(Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1)
-(CSort x0) H7) in (let H9 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead
-(CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e:
-C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c))))
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
-(n0: nat).(eq C c0 (CSort n0))))))) H0 (CSort x0) H7) in (eq_ind_r C (CSort
-x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e)))
-(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
-(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k
-(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort
-n0)))))) (let H10 \def (eq_ind_r T u2 (\lambda (t0: T).((getl n (CHead (CTail
-k u1 c) k0 t) (CHead (CSort x0) (Bind b) t0)) \to (or (ex2 C (\lambda (e:
-C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t)
-(CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen
-c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0))
-(\lambda (n0: nat).(eq C (CSort x0) (CSort n0))))))) H9 u1 H6) in (let H11
-\def (eq_ind_r T u2 (\lambda (t0: T).(getl (r k0 n) (CTail k u1 c) (CHead
-(CSort x0) (Bind b) t0))) H8 u1 H6) in (eq_ind T u1 (\lambda (t0: T).(or (ex2
-C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S
-n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat
-(S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
-nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let
-H12 \def (eq_ind K k (\lambda (k1: K).((getl n (CHead (CTail k1 u1 c) k0 t)
-(CHead (CSort x0) (Bind b) u1)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort
-x0) (CTail k1 u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind
-b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_:
-nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0:
-nat).(eq C (CSort x0) (CSort n0))))))) H10 (Bind b) H5) in (let H13 \def
-(eq_ind K k (\lambda (k1: K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0)
-(Bind b) u1))) H11 (Bind b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or
-(ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e:
-C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
-nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b)))
-(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
-n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e:
-C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n)
-(CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S
-n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_:
-nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1))))))
-(eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C
-(CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t)
-(CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0))
-(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1))
-(\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C
-(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e:
-C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
-nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
-(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
-n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_:
-nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0:
-nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K
-(Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r
-k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) H2)))))) i))))))
-c1)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-inductive cnt: T \to Prop \def
-| cnt_sort: \forall (n: nat).(cnt (TSort n))
-| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt
-(THead k v t))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/cnt/defs.ma".
-
-include "LambdaDelta-1/lift/fwd.ma".
-
-theorem cnt_lift:
- \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i
-d t)))))
-\def
- \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i:
-nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i:
-nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0))
-(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0:
-T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d:
-nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0))
-(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k
-(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csuba/getl.ma".
-
-include "LambdaDelta-1/csuba/props.ma".
-
-include "LambdaDelta-1/arity/props.ma".
-
-include "LambdaDelta-1/csubv/getl.ma".
-
-theorem csuba_arity:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
-t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
-A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
-C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c
-c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall
-(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda
-(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in
-(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda
-(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda
-(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d
-x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc
-g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u
-(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def
-(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl
-i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc
-g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
-a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl
-i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6:
-(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst
-g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
-a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0
-(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u
-(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i)
-x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u
-(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
-(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda
-(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4:
-((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0
-a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2
-u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind
-b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2:
-C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
-a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2)
-\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c
-c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
-(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1:
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3:
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1
-a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1
-(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1:
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g
-a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2:
-C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2
-H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
-(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity
-g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
-C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
-H2)))))))))) c1 t a H))))).
-
-theorem csuba_arity_rev:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
-t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2
-t a))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
-A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0
-a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_:
-(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda
-(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u
-a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to
-(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2
-c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i
-H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
-(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
-u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity
-g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))
-(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x
-(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf
-c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda
-(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2
-(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
-(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let
-H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0
-(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1
-(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono
-c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14
-\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
-with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K
-return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead
-d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0
-(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda
-(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c
-(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda
-(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def
-(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def
-(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16)
-in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13))))))))
-H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
-(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
-u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_:
-(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10:
-(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7
-x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C
-T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda
-(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void)
-x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c
-H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2:
-C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i
-c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2:
-C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c
-(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u)
-(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c
-(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d
-(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2
-(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g
-a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to
-(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2
-c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i
-H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0)
-(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2
-(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst)
-u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4
-Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda
-(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2
-(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
-(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let
-H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0
-(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1
-(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono
-c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14
-\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
-with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K
-return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead
-d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0
-(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda
-(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c
-(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda
-(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def
-(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def
-(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16)
-in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13))))))))
-H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl
-i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let
-H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in
-(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2:
-C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef
-i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda
-(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d
-(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3)
-(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in
-(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
-C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d
-(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2
-(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
-(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0
-a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to
-((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2:
-C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b
-H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c
-H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g
-a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to
-(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
-(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2:
-C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst)
-u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2
-c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3
-(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u)
-(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
-(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda
-(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0
-(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda
-(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4
-H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda
-(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2
-c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0:
-T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g
-c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda
-(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2
-H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
-(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
-C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda
-(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g
-c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2
-H2))))))))))) c1 t a H))))).
-
-theorem arity_appls_appl:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c
-v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t:
-T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs
-(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead
-(Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H:
-(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g
-a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
-TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr)
-v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead
-(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead
-(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B
-Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in
-(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead
-c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u
-t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g
-(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t)
-a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t
-a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H)
-(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst)
-Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1:
-((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
-a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
-Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat
-Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def
-(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2)
-in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c
-(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
-(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0
-x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
-(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3)))))))
-vs))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csuba/defs.ma".
-
-include "LambdaDelta-1/clear/fwd.ma".
-
-theorem csuba_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2))
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1
-c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2))
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
-(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u)
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear
-(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind
-b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2))))
-(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda
-(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g
-c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))))
-(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def
-(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g
-e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1
-e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x:
-C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C
-(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f)
-u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
-C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall
-(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda
-(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3:
-(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1)
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2:
-C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba
-g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b)
-u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2)
-(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3))))))))))))
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_:
-((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2))
-(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda
-(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u
-a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t)
-e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u)
-e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2))
-(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr)
-u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1
-(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
-
-theorem csuba_clear_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1))
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2
-c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear
-c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1))
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
-(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u)
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear
-(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind
-b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2))))
-(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda
-(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g
-c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3))))
-(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def
-(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g
-e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2
-e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x:
-C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C
-(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f)
-u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
-C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall
-(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda
-(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3:
-(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2)
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2:
-C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2:
-C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3
-(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1
-u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3))))))))))))
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_:
-((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1))
-(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda
-(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u
-a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u)
-e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t)
-e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u)))
-(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst)
-t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1
-(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/arity/defs.ma".
-
-inductive csuba (g: G): C \to (C \to Prop) \def
-| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
-| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
-(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
-| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
-(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u:
-T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind
-Abbr) u))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csuba/fwd.ma".
-
-include "LambdaDelta-1/drop/fwd.ma".
-
-theorem csuba_drop_abbr:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
-O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
-c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d1 d2))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g:
-G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))))
-(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1
-(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0:
-(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0
-(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in
-(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba
-g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2
-(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x
-(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda
-(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda
-(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3))))
-H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1:
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g:
-G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
-C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall
-(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S
-n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))
-(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n)
-O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind
-Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3:
-(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n)
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr)
-u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u:
-T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall
-(c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n)
-O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba
-g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda
-(d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c
-(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr)
-u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop
-(r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6:
-(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def
-(csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2:
-C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind
-Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t)
-(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x
-H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
-d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr)
-u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind
-Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
-(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead
-x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) (\lambda (H5:
-(csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O
-c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g c c2 t H5) in
-(let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
-Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2
-(Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2:
-C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind
-Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t)
-(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x
-H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
-d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr)
-u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind
-Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
-(CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead
-x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3
-C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O
-c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0
-(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t
-(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind
-Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u
-H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S
-n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x
-(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat
-(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0
-O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0
-(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7)))))
-(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r
-(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g
-c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda
-(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C
-c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead
-x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def
-(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda
-(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead
-x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal
-nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0:
-nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n
-x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3
-H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t)
-c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let
-H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n)
-O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f)
-x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1)
-(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0
-H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
-d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr)
-u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n)
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
-d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10))))
-H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n
-H1)))))))))))) c1)))) i).
-
-theorem csuba_drop_abst:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i
-O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g:
-G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n
-O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1:
-T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g:
-G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1
-(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl
-c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1
-H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3:
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2:
-C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
-O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x
-(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind
-Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst)
-u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind
-Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
-C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2
-H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind
-Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1
-(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind
-Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr)
-x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind
-Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1)
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6
-H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H:
-((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1
-(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to
-(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall
-(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop
-(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1:
-C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind
-Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0)
-c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O)
-(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda
-(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S
-n) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5)))))
-(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c:
-C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead
-d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to
-(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda
-(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst)
-u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t)
-c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n)
-O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O
-c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
-d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c
-(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst)
-u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop
-(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g
-(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead
-d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7
-\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t)))
-(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr)
-t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t)
-(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2:
-C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
-(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead
-x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def
-(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda
-(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14)
-in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst)
-u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12:
-(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1
-x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0
-x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12
-(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr)
-n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2
-H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda
-(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def
-(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda
-(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g
-c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity
-g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst)
-t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2
-(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x:
-C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c
-x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g
-x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
-Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14:
-(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst)
-u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst)
-n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13:
-(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1
-x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0
-x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13
-(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
-(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst)
-n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2
-H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind
-Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g
-x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1)
-(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda
-(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
-x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
-(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1)
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead
-x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal
-nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0:
-nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in
-(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1)
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x
-(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15:
-(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1
-x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3
-x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4)))
-H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n)
-O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5
-(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17
-H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g
-(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead
-d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7
-\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2:
-T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2:
-C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
-O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
-d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c
-x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g
-x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2)
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1
-(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda
-(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
-(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x:
-C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13:
-(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst)
-u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n
-x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12:
-(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1
-x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3
-x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4)))
-H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n)
-O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5
-(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14
-H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f:
-F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r
-(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c
-c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda
-(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0
-(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f)
-x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2:
-C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda
-(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
-g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10:
-(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1
-x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1)
-H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10:
-(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1
-x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2
-x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f)
-x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind
-Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2
-(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
-
-theorem csuba_drop_abst_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
-O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
-c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g:
-G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n
-O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1:
-C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind
-Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2
-c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1
-(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x
-\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or
-(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C
-(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst)
-u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u)
-(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O
-(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
-d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x
-(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind
-Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x
-(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
-(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void)
-x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1)
-(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O
-(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0
-(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind
-Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H:
-((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1
-(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or
-(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
-C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop
-(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
-(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n)
-O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind
-Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
-(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq
-nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n)
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
-O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u)
-H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u:
-T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall
-(c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop
-(S n) O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda
-(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0:
-K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst)
-u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda (H3:
-(csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c
-(CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c
-(Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to
-(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr)
-t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst)
-u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in
-(or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda
-(d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
-(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t)))
-(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2
-(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
-O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr)
-t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t)
-(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x
-H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind
-Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
-C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
-(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14:
-(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind
-Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr)
-t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
-(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12:
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda
-(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void)
-x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1
-(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12))
-H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc
-g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))
-(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
-A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g
-x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t
-x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C
-(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
-O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda
-(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16:
-(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
-(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind
-Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0
-(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
-x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14))
-(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void)
-x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4
-(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14))
-H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9:
-(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r
-C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2:
-C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let
-H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O
-x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C
-(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
-(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x
-d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
-x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop
-(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda
-(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
-x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12))
-H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst)
-t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst)
-u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in
-(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda
-(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2
-(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba
-g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2
-(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x
-(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H
-c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
-O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
-Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst)
-u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
-O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S
-n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u)
-H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
-Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead
-x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1
-(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12))
-H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9:
-(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r
-C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2:
-C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let
-H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O
-x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C
-(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
-(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x
-d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
-x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop
-(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda
-(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
-x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12))
-H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void)
-t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst)
-u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2:
-C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq
-C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C
-(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S
-n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H
-c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
-O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
-Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst)
-u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
-O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S
-n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u)
-H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
-Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead
-x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1
-(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11))
-H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2
-(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind
-Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def
-H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
-O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0
-(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f)
-x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0
-H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n)
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
-d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0
-(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
-(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat
-f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C
-T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst)
-u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n)
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S
-n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10
-x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
-(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0
-(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda
-(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3
-(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9))
-H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n
-H1)))))))))))) c1)))) i).
-
-theorem csuba_drop_abbr_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i
-O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba
-g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g:
-G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
-(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c:
-C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1
-(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in
-(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2:
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda
-(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
-(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x
-d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C
-(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2:
-C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr)
-u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl
-(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
-A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g
-x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1
-u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C
-(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2:
-C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst)
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2
-(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda
-(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2:
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C
-c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C
-(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop
-O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))
-(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind
-Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T
-(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0
-x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2)))))))))))
-(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall
-(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall
-(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0:
-nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0)
-(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_:
-(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort
-n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1)
-(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let
-H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda
-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
-in (False_ind (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5)))))
-(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c:
-C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead
-d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop
-(S n) O (CHead c k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda
-(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0:
-K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr)
-u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda
-(b: B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop
-(r (Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0:
-B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead
-d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6:
-(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
-(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or3_ind (ex2 C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
-g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
-C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g c t a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-c)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda
-(d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2
-(Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq
-C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C
-(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop
-(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S
-n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
-C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr)
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14:
-(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
-(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda
-(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0
-(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0
-x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n
-x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12:
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void)
-x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind
-Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9))))
-H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind
-Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc
-g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1)
-(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda
-(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
-(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
-C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst)
-x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
-(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x
-d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst)
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0
-(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
-A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16:
-(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18:
-(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
-x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5
-(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17
-H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop
-n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void)
-x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind
-Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2
-H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void)
-x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1)
-(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda
-(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
-(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
-C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void)
-x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
-(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x
-d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0
-(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
-A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14:
-(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16:
-(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
-x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15
-H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop
-n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
-x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind
-Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9)))))
-H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda
-(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
-(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba
-g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8:
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2:
-C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
-Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq
-C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C
-(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop
-(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S
-n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
-C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst)
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14:
-(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
-(Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
-(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda
-(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0
-(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0
-x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n
-x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12:
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void)
-x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
-(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind
-Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9))))
-H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void)
-x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1)
-(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda
-(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
-(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2:
-C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void)
-x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda
-(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x
-d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void)
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0
-(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
-A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14:
-(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16:
-(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
-x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15
-H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop
-n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void)
-x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind
-Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9)))))
-H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda
-(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
-(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda
-(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c))
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x:
-C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x
-c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x
-H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
-(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0
-(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda
-(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
-(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x
-(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))
-x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11))
-(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0
-(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0
-x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n
-x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11:
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void)
-x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x
-(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind
-Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8))))
-H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat
-f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr)
-u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in
-(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda
-(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0
-c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0
-H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda
-(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr)
-u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop
-(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9:
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3:
-T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst)
-x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g
-x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f)
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4
-(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12
-H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0
-(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
-(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2
-(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda
-(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead
-x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0
-(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k
-H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csuba/defs.ma".
-
-theorem csuba_gen_abbr:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
-(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u)
-(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
-C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda
-(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda
-(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C
-c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda
-(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2
-\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2:
-C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
-c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
-(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0)
-(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5
-\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
-u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3)
-in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
-T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr)
-(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C
-c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
-(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0
-c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr)
-u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2
-(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5))
-H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
-c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
-(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1
-(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
-\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr)
-u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2)
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5)))))))))))
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_:
-(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t:
-T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0:
-T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst)
-t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr)
-u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0)
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6))))))))))))
-y c H0))) H))))).
-
-theorem csuba_gen_void:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
-(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2:
-C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void)
-u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda
-(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2)))))
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))
-(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0:
-C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b)
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind
-Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C
-T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2
-(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
-c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u)
-(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match
-e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in ((let H5
-\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
-u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3)
-in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
-T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda
-(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind
-Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda
-(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind
-C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g
-c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2:
-C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b)
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))
-Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6))))
-H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
-c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1
-(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in
-((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead
-c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1
-d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind
-Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3:
-T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2:
-C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C
-c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda
-(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead
-d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba
-g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9)))))
-H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
-c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b)
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc
-g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
-(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C
-(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
-True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1
-(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2:
-C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2)))))
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))
-H6)))))))))))) y c H0))) H))))).
-
-theorem csuba_gen_abst:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
-(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst)
-u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
-C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
-d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y
-c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind
-Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))
-(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst)
-u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C
-(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2))))
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda
-(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2:
-C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
-(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1
-(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
-C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
-\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
-(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
-(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2:
-C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst)
-(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1
-(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2
-d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1
-H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1)
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst)
-u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2
-(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))
-c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5))
-H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
-c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
-C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b
-Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind
-Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind
-Void) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
-Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind
-Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind
-b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3
-C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind
-b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1
-(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc
-g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C
-(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C
-C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t)
-(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t |
-(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind
-Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t
-(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def
-(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let
-H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1))
-\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8)
-in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in
-(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u)
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g
-a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
-a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4))))))))
-H6)))))))))))) y c H0))) H))))).
-
-theorem csuba_gen_flat:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
-(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d1 d2)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead
-d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0:
-(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
-u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u)
-(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
-\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
-u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in
-(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1
-(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2
-k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1
-(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
-(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C
-c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C
-(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1
-(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1
-(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3:
-T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda
-(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat
-f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
-a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
-(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
-(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
-_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
-T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))
-H6)))))))))))) y c H0))) H)))))).
-
-theorem csuba_gen_bind:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
-(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
-(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_:
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y
-c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
-C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
-v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
-(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
-(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
-T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
-(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
-in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
-(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
-b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let
-H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
-(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
-C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b:
-B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1)
-v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
-(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1
-(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void
-b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c:
-C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1
-H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9)
-in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind
-b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b)
-u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2))
-H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
-(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3
-B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
-e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t
-(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5:
-(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind
-Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind
-Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda
-(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t
-| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind
-b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1
-e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a)))
-H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc
-g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let
-H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15
-\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to
-(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
-e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7))
-H6)))))))))))) y c2 H0))) H)))))).
-
-theorem csuba_gen_abst_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
-(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u)
-(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
-C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y:
-C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1:
-C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C
-c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n:
-nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def
-(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda
-(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
-d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
-H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
-c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C
-(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k
-u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
-(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3)
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1
-(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C
-c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C
-(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))
-(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C
-(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let
-H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to
-(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g
-c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind
-Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
-(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind
-Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda
-(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1
-(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not
-(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead
-c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2
-| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind
-Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1
-(Bind Abst) u) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
-\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in
-(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def
-(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11
-\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to
-(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1
-(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g
-c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind
-Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1)
-(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1
-(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1))
-H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
-(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or
-(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc
-g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
-(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C
-(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
-False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead
-d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead
-c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind
-Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d2 d1))))) H6)))))))))))) c y H0))) H))))).
-
-theorem csuba_gen_void_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
-(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
-Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u)
-(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
-C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda
-(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda
-(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
-c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda
-(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2
-\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2:
-C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
-d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
-c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
-(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
-d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0)
-(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5
-\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
-u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3)
-in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
-T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void)
-(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C
-c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
-(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
-d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1
-c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void)
-u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1
-(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5))
-H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
-c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
-(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
-d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1
-(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in
-((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
-C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in
-((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead
-c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b
-Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0:
-B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda
-(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
-c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9)
-in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in
-(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_:
-False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13)))))))
-H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
-c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
-(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
-d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
-a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
-(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C
-(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
-return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
-False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead
-d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1
-(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))
-H6)))))))))))) c y H0))) H))))).
-
-theorem csuba_gen_abbr_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c
-(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr)
-u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda
-(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda
-(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C
-(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
-C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind
-Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2
-C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda
-(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C
-c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
-(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match
-e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5
-\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
-u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
-in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
-T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t)
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2:
-C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
-C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C
-c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C
-(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
-C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba
-g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1
-(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
-(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
-(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
-(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b:
-B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2:
-T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr)
-u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b)
-u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda
-(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
-Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0
-(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
-u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void)
-u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in
-(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in
-(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0)
-(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a:
-A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq
-C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2:
-C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void)
-u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C
-(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
-(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t:
-T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u:
-T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr)
-u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
-(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind
-Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0)
-\Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in
-(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity
-g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g
-c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0
-(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g
-a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1
-a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8)
-in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in
-(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
-(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1
-u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10))))))))
-H6)))))))))))) c y H0))) H))))).
-
-theorem csuba_gen_flat_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
-(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead
-d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0:
-(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
-u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
-(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
-\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
-u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in
-(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1
-(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1
-k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2
-(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
-(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C
-c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C
-(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
-C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1
-(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1
-(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3:
-T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda
-(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat
-f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
-a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
-(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
-(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
-_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
-T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
-H6)))))))))))) c y H0))) H)))))).
-
-theorem csuba_gen_bind_rev:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
-(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C
-(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_:
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2
-y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq
-C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
-v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda
-(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1
-(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u:
-T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u)
-(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3)
-in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
-(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
-b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let
-H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1
-(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
-C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b:
-B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let
-H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3
-(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B
-(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3
-(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) u2) (CHead
-e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3
-e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1
-H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C
-c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1))
-H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
-(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3
-B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
-e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t
-(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5:
-(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind
-Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind
-Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda
-(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u
-| (CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind
-b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3
-e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8)
-in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10)
-in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1)
-v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
-C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
-(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3
-(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1
-(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))
-Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7))
-H6)))))))))))) c2 y H0))) H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csuba/drop.ma".
-
-include "LambdaDelta-1/csuba/clear.ma".
-
-include "LambdaDelta-1/getl/clear.ma".
-
-theorem csuba_getl_abbr:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
-(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g
-c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d1 d2))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e:
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u)))
-(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x:
-C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
-Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1
-(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda
-(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1
-(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
-d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3:
-(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1
-(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to
-((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2:
-C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda
-(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0
-(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
-(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
-Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u)
-t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda
-(c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda
-(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
-(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr
-H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c
-(Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15
-g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1:
-C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18:
-(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1
-(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18))))
-H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead
-x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind
-Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c
-(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n
-O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead
-x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10
-\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t)
-(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0
-(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6)
-f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead
-d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1
-(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))
-(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u)
-x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u
-H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))
-(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda
-(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2
-c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl
-O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3
-(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16)))))
-H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1:
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2)
-\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda
-(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O
-x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1
-c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B
-C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind
-b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead
-x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3:
-C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2)
-x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def
-(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C
-(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2:
-C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15:
-(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x
-\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B
-C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
-x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda
-(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19:
-(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16
-(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind
-C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22:
-(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1
-x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead
-x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11))))))))
-i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
-
-theorem csuba_getl_abst:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall
-(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e:
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1)))
-(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear
-x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to
-((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2)
-\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda
-(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n)
-(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4
-(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0
-(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2
-C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda
-(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear
-(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O
-c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1))
-\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i
-c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b)
-t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst)
-u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c]))
-(CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead
-d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst |
-(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind
-Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst)
-u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0)
-\Rightarrow t0])) (CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B
-Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba
-g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0
-(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0:
-B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in (let H15 \def
-(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1
-H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind
-(ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda
-(d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst)
-u1))).(\lambda (H19: (csuba g d1 x1)).(or_introl (ex2 C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2
-(CHead x1 (Bind Abst) u1) (CHead x1 (Bind Abst) u1) H18 (clear_bind Abst x1
-u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1:
-C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1
-(Bind Abbr) x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1
-u1 (asucc g x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))
-x1 x2 x3 (getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2)
-H18 (clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8))
-H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f)
-t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst)
-u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0
-(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
-c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1:
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2)
-\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda
-(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
-C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c:
-C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat
-f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1)
-(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def
-(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1)
-H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2))
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1)
-x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1
-H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
-(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3
-(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2
-(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in
-(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2
-(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
-A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba
-g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity
-g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13
-(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O
-c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead
-x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15))
-H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1:
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2)
-\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat
-f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def
-(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b:
-B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v)))))
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat
-f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12:
-(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0
-(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2)
-x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2))
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4)
-x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5
-x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda
-(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda
-(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl
-(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7:
-C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6)
-x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c:
-C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13
-x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or
-(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
-(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda
-(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2:
-C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
-n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7
-(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C
-(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
-C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23)
-H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
-u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst)
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23:
-(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1
-x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9
-x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6
-c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22))
-H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1
-H2)))) H0))))))).
-
-theorem csuba_getl_abst_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
-(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g
-c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e:
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u)))
-(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x:
-C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
-Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1
-(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))
-(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear
-(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u)
-n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl
-i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0:
-C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u))
-\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4:
-(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0:
-K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind
-Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b:
-B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear
-(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u)
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in
-((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
-C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0
-in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
-Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u)
-t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda
-(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda
-(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
-(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst
-H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c
-(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u
-H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C
-(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1:
-C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19:
-(csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1
-(getl_intro i c2 (CHead x1 (Bind Abst) u) (CHead x1 (Bind Abst) u) H18
-(clear_bind Abst x1 u)) H19))))) H17)) (\lambda (H17: (ex2_2 C T (\lambda
-(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2:
-C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Void)
-x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 x2 (getl_intro i c2
-(CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 (clear_bind Void x1
-x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5:
-(drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f)
-t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c:
-C).((drop i O c (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c)
-\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop
-n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or
-(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n c2
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f)
-t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C
-x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1
-(CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind
-Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) f t) in (let H11
-\def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst)
-u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abst) u)))
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead
-d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda
-(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (H12:
-(csuba g x2 (CHead d1 (Bind Abst) u))).(\lambda (H13: (clear c2 x2)).(let H_x
-\def (csuba_gen_abst_rev g d1 x2 u H12) in (let H14 \def H_x in (or_ind (ex2
-C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead
-d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda
-(d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
-(or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl
-O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g
-d2 d1))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst)
-u))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c:
-C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u) H16) in (or_introl (ex2 C
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst)
-u) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda
-(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl O c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3:
-C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind Void)
-x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c:
-C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in (or_intror (ex2 C
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3
-x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 (drop_refl c2) H18)
-H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8:
-((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2:
-C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9:
-(drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10:
-(csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in
-(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1
-(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
-T).(drop n O e (CHead x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S
-n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C
-T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2:
-B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind
-x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def
-(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C
-(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2:
-C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: C).(\lambda (H15: (csuba
-g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def
-(csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C
-T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8:
-T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g
-x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead
-x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C
-(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
-(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (H23: (getl
-n x7 (CHead x9 (Bind Abst) u))).(\lambda (H24: (csuba g x9 d1)).(or_introl
-(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl
-(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8
-H20 (CHead x9 (Bind Abst) u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
-(d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl
-(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9:
-C).(\lambda (x10: T).(\lambda (H23: (getl n x7 (CHead x9 (Bind Void)
-x10))).(\lambda (H24: (csuba g x9 d1)).(or_intror (ex2 C (\lambda (d2:
-C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
-(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10
-(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24))))))
-H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4)))))))
-x H1 H2)))) H0))))))).
-
-theorem csuba_getl_abbr_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall
-(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba
-g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e:
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1)))
-(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))
-(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead
-d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c
-(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3
-(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_:
-(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr)
-u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba
-g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0:
-C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1))
-\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl
-i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
-d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1
-(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr)
-u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear
-(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2
-c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b:
-B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear
-(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u1)
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6))
-in ((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
-(_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow
-(match k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
-(Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b)
-t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
-Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr)
-u1) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1
-x0)).(\lambda (c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r
-T t (\lambda (t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let
-H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0)
-u1))) H13 Abbr H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O
-c1 (CHead c (Bind Abbr) u1))) H14 d1 H11) in (let H16 \def
-(csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in (or3_ind (ex2 C (\lambda
-(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C
-(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1
-(Bind Abbr) u1))).(\lambda (H19: (csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2
-(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1
-u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1:
-C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1
-(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1
-x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3
-(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18
-(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
-(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl
-i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
-(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind
-Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead
-x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16))))))))))
-H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f)
-t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr)
-u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop
-n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3
-(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8:
-(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g
-c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead
-x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def
-(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind
-Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f)
-t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2:
-C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2))
-(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2:
-C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13:
-(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14
-\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda
-(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3:
-C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba
-g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead
-x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
-(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2
-(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
-A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba
-g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity
-g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13
-(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst)
-x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda
-(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind
-Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2
-(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in
-(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda
-(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3
-(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11))))))))
-(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda
-(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1
-(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let
-H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T
-(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b)
-v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0
-(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3
-(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14
-\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind
-C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2:
-C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16:
-(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let
-H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S
-n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8:
-T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g
-x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead
-x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C
-(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S
-n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C
-(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda
-(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24:
-(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C
-(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr)
-u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2:
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11:
-A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24:
-(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda
-(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20
-(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22:
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T
-(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2))))
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2:
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_:
-T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23:
-(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9
-d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro
-C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10
-(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24))))))
-H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4)))))))
-x H1 H2)))) H0))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csuba/defs.ma".
-
-theorem csuba_refl:
- \forall (g: G).(\forall (c: C).(csuba g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0))
-(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0
-c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/csuba.ma".
-
-theorem csubc_arity_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to
-(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
-c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t
-a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))).
-
-theorem csubc_arity_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to
-((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to
-(arity g c1 t a))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
-c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda
-(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2
-H) H0)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/fwd.ma".
-
-theorem csubc_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall
-(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda
-(e2: C).(csubc g e1 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1
-e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c
-c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0
-e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2:
-C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def
-(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2
-C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g
-e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
-(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq
-C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda
-(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3
-g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
-a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda
-(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2))
-(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C
-(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e
-c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda
-(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
-C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2
-(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x
-(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda
-(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2:
-C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind
-b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind
-b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda
-(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
-C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
-(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda
-(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind
-b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda
-(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7:
-(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2
-C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b)
-u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b]))
-(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda
-(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g
-(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0
-(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2))
-(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2
-H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda
-(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0)
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind
-Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e
-c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda
-(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2))
-(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0:
-B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind
-x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B
-x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2)
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc
-g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0:
-K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
-(Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void
-(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2))
-(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda
-(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead
-e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2)
-(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1))))))))
-(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
-((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2))
-(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u:
-T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x
-\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind
-(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3:
-C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
-A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda
-(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
-(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3:
-C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2:
-C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C
-(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e
-c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda
-(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
-C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f)
-u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda
-(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c
-e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda
-(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2:
-C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2)))
-(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c
-x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda
-(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5))))
-H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
-A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda
-(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
-(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda
-(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
-C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
-(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2
-e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6:
-(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda
-(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C
-(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0
-e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f)
-(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in
-(False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2))
-(\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4:
-(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2
-(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
-T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3:
-C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2:
-C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda
-(x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0)
-x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0
-Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2)
-(\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2:
-C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match
-ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat
-_) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2:
-C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9))
-c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/defs.ma".
-
-include "LambdaDelta-1/sc3/props.ma".
-
-theorem csubc_csuba:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba
-g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
-c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda
-(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
-(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda
-(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b:
-B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
-T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v:
-T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w:
-T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g
-c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sc3/defs.ma".
-
-inductive csubc (g: G): C \to (C \to Prop) \def
-| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
-| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
-(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
-| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
-(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g
-a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr)
-w))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/fwd.ma".
-
-include "LambdaDelta-1/sc3/props.ma".
-
-theorem csubc_drop_conf_O:
- \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h
-O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2:
-C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1:
-C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2)
-\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1
-e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H:
-(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n)
-c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda
-(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1:
-(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O
-O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2
-e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c:
-C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c
-e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2:
-C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2))))
-(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1:
-C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2)
-\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1
-e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h:
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall
-(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2
-e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c
-k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind
-C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2))
-(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O
-c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1
-(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0:
-(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t)
-c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g
-e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2:
-C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l
-g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C
-c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda
-(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T
-(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b)
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3)))))
-(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1
-e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t)))
-(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2
-(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2:
-C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x:
-C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c
-x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop
-(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k
-n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C
-(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2))
-(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2:
-C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x
-x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n)
-O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0
-H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c
-t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2
-C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k
-(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda
-(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3
-g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C
-(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2))))
-(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1))
-(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k
-(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g
-(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda
-(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r
-(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2:
-C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2:
-C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1
-e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g
-e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1)
-e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13
-x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda
-(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2)))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void)))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
-(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c
-c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2
-e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda
-(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8:
-(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C
-(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2))))
-(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1))
-(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k
-(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g
-(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda
-(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r
-(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2:
-C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2:
-C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1
-e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g
-e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2)
-e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2)
-H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)).
-
-theorem drop_csubc_trans:
- \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
-(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
-C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
-(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
-(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
-(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat
-h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
-C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
-(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
-nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g
-(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
-C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def
-(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C
-(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1))
-e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
-(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
-nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c
-c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
-e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h
-n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h:
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
-(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
-(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O
-(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2
-\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t)
-(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
-O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1)
-H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
-(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1
-e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop
-(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2
-e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
-(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
-(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
-e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda
-(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C
-(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k
-t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t)))))
-H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
-(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda
-(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t)
-c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
-e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda
-(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
-T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
-n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
-C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
-x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
-(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
-(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to
-(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
-e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1)
-H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0
-n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k
-x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1:
-C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r
-T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n)
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def
-(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C
-(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0
-c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
-(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1
-(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
-(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
-a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n)
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))
-(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda
-(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k
-x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n)
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))
-(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12:
-(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda
-(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r
-k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def
-H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1:
-C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1)))
-(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2:
-C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c
-x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
-(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r
-k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k
-n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0
-x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))
-(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g
-(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3:
-T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C
-e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14:
-(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C
-(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S
-n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))))
-(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n
-(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3:
-C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
-e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1))))))))
-H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r
-k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0:
-K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
-(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0
-\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind
-C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C
-(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1:
-C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1)))
-(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c
-x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr)
-x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst)
-n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19
-Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g
-(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g
-x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda
-(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1
-(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
-T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3:
-C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1:
-C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n)
-x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11:
-(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind
-Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0
-x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda
-(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r
-k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0:
-nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to
-(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1:
-C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n)
-x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda
-(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind
-Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3
-(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1))
-c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def
-H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc
-g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4)))
-(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1))
-c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g
-c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2)
-x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void)
-n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18
-x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n
-x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4)))))))))
-(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
-
-theorem csubc_drop_conf_rev:
- \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
-(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
-C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
-(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
-(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
-(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat
-h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
-C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
-(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
-nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1
-(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
-C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def
-(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C
-(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n)))
-e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
-(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
-nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1
-c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
-e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h
-n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h:
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
-(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
-(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O
-(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2
-\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t)
-(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
-O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1)
-H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
-(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1
-e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop
-(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1
-e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
-(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
-(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
-e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda
-(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C
-(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c
-k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t)))))
-H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
-(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda
-(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k
-t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
-e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda
-(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
-T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
-n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
-C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
-x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
-(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
-(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to
-(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
-e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1)
-H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0
-n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0
-k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc
-g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h
-(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1))
-(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def
-(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C
-(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
-x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
-(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1
-(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g
-(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a
-x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq
-C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_:
-T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C
-(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
-x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1:
-C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x:
-C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x
-x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1:
-C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k
-n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in
-(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g
-c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2:
-C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2
-c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r
-k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k
-n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1:
-C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v)))))
-(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda
-(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
-(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind
-Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
-x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2
-C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead
-c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
-A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2
-(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g
-(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2
-(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1
-c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let
-H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c
-k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3
-(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda
-(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr)
-H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5
-(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda
-(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc
-g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind
-Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1:
-C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1:
-C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1
-(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x:
-C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C
-(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1:
-C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x
-(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst
-g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19)
-(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n)
-H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T
-(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
-(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1
-x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1:
-T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1:
-C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n)
-c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))
-(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1
-(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13:
-(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3
-(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1
-c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let
-H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c
-k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3
-(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda
-(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2)
-H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5
-(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1:
-C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1
-(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h
-H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1
-x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n)
-c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind
-x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n
-x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S
-n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind
-x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4))
-(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n
-x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10))
-H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/drop.ma".
-
-theorem drop1_csubc_trans:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
-(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2
-e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
-c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def
-(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c:
-C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1
-e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda
-(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2:
-C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1)
-\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
-c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n
-n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def
-(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
-(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda
-(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))
-(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x
-e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C
-(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C
-(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2
-c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g
-x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def
-H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1:
-C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1))
-(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0
-x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1
-(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0
-n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
-
-theorem csubc_drop1_conf_rev:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
-(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1
-e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
-c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def
-(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c:
-C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1
-e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda
-(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2:
-C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2)
-\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
-c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n
-n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def
-(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
-(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda
-(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))
-(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x
-e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C
-(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C
-(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1
-c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g
-x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8
-\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1:
-C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1))
-(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0
-x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1
-(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0
-n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/defs.ma".
-
-theorem csubc_gen_sort_l:
- \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to
-(eq C x (CSort n)))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g
-(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda
-(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c))))
-(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def
-(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with
-[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0)
-(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort
-n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2:
-C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C
-c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v)
-(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
-_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v)
-(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
-(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
-c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort
-n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (ee: C).(match
-ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
-(CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead
-c2 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1
-(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_:
-(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2
-w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def
-(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
-\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr)
-w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))).
-
-theorem csubc_gen_head_l:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k:
-K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x
-(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2:
-C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w)))))
-(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
-(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T
-(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b)
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1
-c2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k:
-K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v)
-(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2:
-C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
-(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind
-Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
-c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
-v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))
-(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead
-c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k
-(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1
-c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda
-(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda
-(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C
-T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
-(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind
-Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
-c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
-v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))
-(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0
-(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_:
-T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n)
-(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _
-_ _) \Rightarrow False])) I (CHead c1 k v) H1) in (False_ind (or3 (ex2 C
-(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g
-c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
-(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort
-n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
-(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
-a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2:
-T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2:
-C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2:
-C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v))
-\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3:
-C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
-A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w:
-T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda
-(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0:
-K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k
-v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
-(CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K (\lambda
-(e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0
-| (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in
-((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead
-c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq
-C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C
-(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C
-T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead
-c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc
-g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g
-a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3
-w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
-(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3
-(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3:
-C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
-A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda
-(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3
-(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k
-(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
-c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v))
-\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3:
-C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
-A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w:
-T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda
-(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let
-H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in
-(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v)))
-(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
-T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda
-(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda
-(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind
-b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
-c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v)))
-(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0
-H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda
-(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2
-C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1
-c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
-(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2
-(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
-(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
-a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not
-(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead
-c0 (Bind Void) u1) (CHead c1 k v))).(let H5 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
-(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4)
-in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _)
-\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in ((let H7
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0
-(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void)
-k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c:
-C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead
-c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T
-(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind
-b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
-c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c
-c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1
-(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v)))
-(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
-T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda
-(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0)
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind
-Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
-c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3
-(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v)))
-(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
-T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T
-(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b)
-u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_:
-T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
-T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2
-(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3)))
-(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind
-Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C
-(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda
-(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w:
-T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda
-(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0)
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void)
-(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B
-b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1
-c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda
-(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C
-(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6))
-H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0
-c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3:
-C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
-v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))
-(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2
-(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
-T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3:
-(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2
-w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0
-(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind
-Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1
-k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow
-t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K
-(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0
-(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C
-c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def
-(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C
-(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3)))
-(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind
-Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead
-c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g
-(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3
-g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind
-C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K
-k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3:
-C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst)))))
-(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
-Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0)
-c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3
-w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
-c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
-T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst)
-(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w)
-(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda
-(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3:
-C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3
-(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g
-c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g
-a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3
-w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
-(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3:
-C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3:
-C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0:
-T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr)
-w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v))))
-(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))))
-(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead
-c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda
-(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w)
-(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g
-(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3
-g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2
-(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0)))
-H)))))).
-
-theorem csubc_gen_sort_r:
- \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to
-(eq C x (CSort n)))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x
-(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda
-(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g
-(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0))))
-(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def
-(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with
-[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0)
-(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort
-n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2:
-C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C
-c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v)
-(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
-_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v)
-(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
-(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1
-c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort
-n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
-_ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1
-(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2
-(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_:
-(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2
-w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def
-(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
-\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst)
-v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))).
-
-theorem csubc_gen_head_r:
- \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k:
-K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x
-(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1:
-C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v)))))
-(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda
-(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T
-(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
-(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2)))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k:
-K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w)
-(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1:
-C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
-(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind
-Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
-c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))
-(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead
-c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K
-k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1
-c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda
-(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda
-(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C
-T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
-(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind
-Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
-c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))
-(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead
-c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K
-k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1
-c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k
-w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda
-(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2)))
-(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind
-Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n)
-(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g
-(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a
-c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq
-C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1:
-C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0:
-C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w))
-\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3:
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_:
-A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v:
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda
-(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0:
-K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(let
-H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0
-v) (CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e
-in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1
-_) \Rightarrow k1])) (CHead c0 k0 v) (CHead c2 k w) H3) in ((let H6 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v)
-(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0
-c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead
-c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t)
-(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3
-g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3
-g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1:
-T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k
-(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3
-k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3:
-C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind
-Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3
-c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a)
-c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))
-(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead
-c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda
-(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1
-(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3:
-C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda
-(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
-T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void)))))
-(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2
-H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8)
-in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w)))
-(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0:
-T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda
-(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
-T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w)
-(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
-T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not
-(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g
-c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k
-w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w))
-H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0:
-C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w))
-\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3:
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_:
-A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v:
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda
-(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b:
-B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b)
-u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e
-in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead
-_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let
-H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0
-(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda
-(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead
-c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda
-(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
-T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3:
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3:
-C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda
-(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void)
-v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind
-b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3
-c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g
-c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2
-(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w)))
-(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
-T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3:
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3:
-C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda
-(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void)
-v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind
-b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3
-c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2
-C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda
-(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst)
-v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
-(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
-T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind
-Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3:
-C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3:
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v:
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst)
-v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
-(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C
-T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind
-Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_:
-C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_:
-B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead
-c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K
-(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not
-(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g
-c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K
-(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda
-(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k
-w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3:
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_:
-A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v:
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda
-(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_:
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v:
-T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0:
-T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr)
-w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
-\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7
-\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0)
-(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq
-C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8)
-in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in
-(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3
-(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g
-c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
-(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1
-(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3
-g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0:
-A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda
-(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind
-C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K
-k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3:
-C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr)))))
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind
-Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3
-c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0)
-c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2
-w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C
-c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_:
-T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not
-(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g
-c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0:
-K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0
-w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda
-(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda
-(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst)
-v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3
-v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))
-(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead
-c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3:
-C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3:
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
-(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0:
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst)
-v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2))))
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3
-v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))
-(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead
-c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda
-(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v)
-(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
-A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3
-g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0:
-A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead
-c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0)))
-H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/drop.ma".
-
-include "LambdaDelta-1/csubc/clear.ma".
-
-theorem csubc_getl_conf:
- \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i
-c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2:
-C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda
-(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1
-\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e))
-(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2))
-(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1
-x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2
-c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2))
-(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2))
-(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O
-c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1
-H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2))
-(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2))
-(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0
-x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2
-e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8)
-H9)))) H7)))))) H4)))))) H1)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/defs.ma".
-
-include "LambdaDelta-1/sc3/props.ma".
-
-theorem csubc_refl:
- \forall (g: G).(\forall (c: C).(csubc g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0))
-(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0
-c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst0/props.ma".
-
-include "LambdaDelta-1/csubst0/fwd.ma".
-
-include "LambdaDelta-1/clear/fwd.ma".
-
-theorem csubst0_clear_O:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
-(\forall (c: C).((clear c1 c) \to (clear c2 c))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
-T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2
-c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
-(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n)
-c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda
-(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0:
-C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t)
-c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
-T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
-nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
-nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
-t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat
-(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
-(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
-c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t
-u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
-(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0:
-T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C
-c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k
-x0) (\lambda (c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c
-k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0))))
-(\lambda (b: B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
-(eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind
-b) x0) c0) H8))))) (\lambda (f: F).(\lambda (H6: (clear (CHead c (Flat f) t)
-c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1
-(\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0
-(clear_gen_flat f c c0 t H6) f x0))))) k H1 H3) c2 H4)))))) H2)) (\lambda
-(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))
-(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
-v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq
-nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5:
-(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3
-c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0
-x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear
-(CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8
-\def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7)
-in (False_ind (clear (CHead x0 (Bind b) t) c0) H8))))) (\lambda (f:
-F).(\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s
-(Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c
-x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t
-H6)) f t))))) k H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda
-(_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
-(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
-nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k
-x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c
-x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) (K_ind
-(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to
-(clear (CHead x1 k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear (CHead c
-(Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def
-(eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8)
-in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9))))) (\lambda (f:
-F).(\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s
-(Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
-x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v
-t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t
-H7)) f x0)))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O
-H0))))))))))) c1).
-
-theorem csubst0_clear_O_back:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
-(\forall (c: C).((clear c2 c) \to (clear c1 c))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
-T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c
-c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
-(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2
-c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c:
-C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to
-(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O
-v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda
-(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
-T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
-v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
-nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
-c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear
-(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O
-(s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v
-t x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead c
-k x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead
-c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7:
-(eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0)
-c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda
-(f: F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead
-c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n
-v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t)))))
-k H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
-c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0)
-(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k
-x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c
-x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k
-t) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0
-k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: (eq
-nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let
-H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7)
-in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda (f:
-F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0
-(Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v
-c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t
-H8)) f t))))) k H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
-(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda
-(x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1
-k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c
-x1)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x1 k
-x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead
-x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8:
-(eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0)
-c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda
-(f: F).(\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead
-x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n:
-nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda
-(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0
-(clear_gen_flat f x1 c0 x0 H9)) f t)))))) k H3 H7))))))))) H2))
-(csubst0_gen_head k c c2 t v O H0))))))))))) c1).
-
-theorem csubst0_clear_S:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0
-(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4
-B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq
-C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v e1 e2))))))))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
-T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c
-c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c:
-C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4
-(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))))))))
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i:
-nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4
-(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t)
-c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
-T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda
-(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
-T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
-v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
-(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t
-u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
-(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k
-x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda
-(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat
-(S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b)
-u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))
-(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
-(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e:
-nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S
-n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1
-(\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind
-b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T
-(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
-(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b)
-x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda
-(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq
-C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead
-e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b)
-t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b)
-x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
-t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
-(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1))))))
-(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0
-(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0
-(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear
-(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let
-H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in
-(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i)
-H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))
-(clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))) k H1 H3) c2
-H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2
-(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1:
-nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0
-k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda
-(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat
-(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b)
-u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))
-(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
-(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e:
-nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S
-n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1
-(\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind
-b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T
-(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
-(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b)
-t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
-(CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t))
-(ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b)
-t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
-t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
-(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u))))))
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
-(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t
-(refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0
-(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear
-(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let
-H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in
-(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i)
-H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in
-(or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat
-f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
-t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t)))
-(\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t)
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3:
-C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind
-x2) x4))).(\lambda (H13: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H14:
-(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
-t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
-t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0
-(CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
-t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0
-(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2
-(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t)
-H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2)
-x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14:
-(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear
-(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0
-(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))) k H1 H3)
-c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda
-(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda
-(H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda
-(H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4
-(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind
-(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to
-(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda
-(H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b)
-x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
-(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i)
-(S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
-x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v
-t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4
-(clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1))))))
-(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
-T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
-(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1)))))))
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3
-(clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda
-(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind
-b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0)
-u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0)
-u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))
-(ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
-(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0)
-H11 H10)) c0 (clear_gen_bind b c c0 t H7)))))))) (\lambda (f: F).(\lambda
-(H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f)
-x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f)
-x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
-x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0
-n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c
-c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat
-f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f)
-x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0)))
-(\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0)
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4:
-C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind
-x3) x5))).(\lambda (H15: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H16:
-(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f)
-x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f)
-x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1
-(CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B
-C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
-c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f)
-x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
-T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1
-(CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2
-(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0)
-H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
-T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3)
-x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16:
-(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear
-(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1
-(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1
-H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1).
-
-theorem csubst0_clear_trans:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0
-i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C
-(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or
-(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1:
-C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1
-u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2)
-e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear
-(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda
-(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2)
-(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1:
-C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind
-b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2))
-(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2)))
-(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda
-(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear
-(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1
-u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda
-(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear
-(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2
-(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k:
-K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
-T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2:
-C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0
-v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda
-(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0:
-K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C
-(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead
-c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u)
-e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3
-(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c))
-(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear
-(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0
-(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind
-b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4
-(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3
-(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2
-(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead
-c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in
-(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0
-v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u)
-e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear
-(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear
-(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f
-u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda
-(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
-(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x
-e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2)
-(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead
-c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f
-u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0:
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0
-i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0
-c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2)
-(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3
-e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind
-(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1)
-e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1:
-C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear
-(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c:
-C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s
-(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1)))))
-(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C
-(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1:
-C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0
-(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind
-b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4
-H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2
-(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2)
-(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3
-e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0
-i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda
-(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
-(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda
-(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or
-(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1
-e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x:
-C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror
-(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1
-e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
-(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k
-H3))))))))))))) i v c1 c2 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/defs.ma".
-
-include "LambdaDelta-1/C/defs.ma".
-
-inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
-| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1:
-T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i)
-v (CHead c k u1) (CHead c k u2))))))))
-| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s
-k i) v (CHead c1 k u) (CHead c2 k u))))))))
-| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall
-(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall
-(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2
-k u2)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst0/fwd.ma".
-
-include "LambdaDelta-1/drop/fwd.ma".
-
-include "LambdaDelta-1/s/props.ma".
-
-theorem csubst0_drop_gt:
- \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
-c1 e) \to (drop n O c2 e)))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
-\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda
-(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
-O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H:
-((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
-(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop
-n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
-(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
-c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))
-(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v
-(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1)
-e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0)
-O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0)
-O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop
-(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee
-in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6))
-e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda
-(H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e:
-C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i
-v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k
-t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
-(u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
-v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat
-(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
-nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
-c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S
-n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k
-x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t
-x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let
-H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
-T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop
-(S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda
-(n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop
-(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1)
-v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
-e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0)
-e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda
-(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
-(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
-e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c
-e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c
-e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1)
-v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
-e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O)
-(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
-(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1
-O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m:
-nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
-(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
-(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
-x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13))
-(lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2
-H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
-c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda
-(x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6:
-(eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C
-(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind
-nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
-c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
-e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1:
-nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0
-n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c
-c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
-e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t)
-e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda
-(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to
-(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
-e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0
-x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f:
-F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3:
-C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt
-(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq
-nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat
-f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7
-e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
-(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
-m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e)
-(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
-n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS
-x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4))
-(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0:
-T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k
-x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
-x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda
-(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1:
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
-(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5)
-in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
-H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3:
-C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop
-(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to
-(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop
-(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0:
-T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c
-e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S
-n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11)
-x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda
-(H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3)
-\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3
-e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O)
-(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
-(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2
-O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2
-nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m
-n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m:
-nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x:
-nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat
-f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k
-(drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head
-k c c2 t v i H2))))))))))) c1)))))) n).
-
-theorem csubst0_drop_gt_back:
- \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
-c2 e) \to (drop n O c1 e)))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
-\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda
-(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
-O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H:
-((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall
-(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop
-n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda
-(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
-c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))
-(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i
-v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2
-e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e))))))))
-(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v
-c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c
-e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v:
-T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
-(H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
-c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C
-nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e)
-(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
-(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k
-x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda
-(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind
-nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
-c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c
-e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1:
-nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall
-(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1)
-(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e)))))
-(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
-(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
-(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda
-(H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t)))))
-(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s
-(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop
-(S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda
-(H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda
-(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O
-(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c
-e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
-(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
-m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e)
-(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
-n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k
-H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C
-nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
-c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S
-n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq
-nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7:
-(csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0)
-O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1:
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
-(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5)
-in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1)
-H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
-(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
-n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to
-(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt
-(s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0
-e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13)
-t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0:
-T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
-e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S
-n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O)
-(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
-(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop
-(Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m:
-nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
-(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
-(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
-x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13)
-t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0
-H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O
-(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
-nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k
-x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c
-x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3
-(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1:
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
-(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5)
-in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2)
-H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0
-(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S
-n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to
-(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt
-(s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1
-e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14)
-t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0:
-T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3
-e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S
-n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O)
-(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
-(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop
-(Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m:
-nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
-(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
-(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S
-x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14)
-t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0
-H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
-
-theorem csubst0_drop_lt:
- \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
-c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k
-w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w)))))))
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
-n)) v e1 e2))))))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
-\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T
-T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
-O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i:
-nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v:
-T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O
-c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
-c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0:
-nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c)
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
-c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k:
-K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r
-nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O))
-(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1))
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1)
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
-k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1)
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
-T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C
-(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k:
-K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
-T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0
-O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0
-O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda
-(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0))
-in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda
-(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s
-k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead
-e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
-C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
-(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0
-(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in
-(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k
-u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k
-i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
-u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
-T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
-v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0
-u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k
-i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0
-u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
-T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O))
-v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k
-u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1
-u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3
-c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0
-O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1
-e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1
-u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r
-nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O))
-(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1))
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k
-u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
-k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1)
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
-T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k
-u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u
-w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4
-u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5
-H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda
-(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1:
-C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e:
-C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))
-(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead
-e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H:
-(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
-C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c
-e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
-e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1:
-nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1)
-c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind
-(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e)
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S
-n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq
-nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c:
-C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
-e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind
-nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop)
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind
-(or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u))))))
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
-O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort
-n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1
-k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
-e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c:
-C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
-(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C
-T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
-e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
-(H2: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
-n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))
-(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
-(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4:
-(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_:
-(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
-n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
-H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
-T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
-(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
-(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
-T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
-(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
-(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i
-(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1)
-(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
-(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
-(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
-(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
-(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
-u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
-e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c
-k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead
-e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u))))))
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
-k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
-(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b)
-x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
-b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
-e2))))))) (drop_drop (Bind b) n0 c e H9 x0)))))) (\lambda (f: F).(\lambda
-(H9: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall
-(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0)
-O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
-x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
-f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2))))))) (drop_drop (Flat f) n0 c e H9 x0)))))) k (drop_gen_drop k c e t n0
-H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
-v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4:
-(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6:
-(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop
-(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let
-H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0:
-T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4
-(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3
-(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C
-T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0
-(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
-(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i
-(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1)
-(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
-(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
-(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
-(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1)
-(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2))))))
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0
-u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1
-e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0
-k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead
-e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u))))))
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
-k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w))))))
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
-(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b)
-x1))).(let H12 \def (IHn x1 (le_S_n (S n0) x1 H11) c x0 v H6 e H9) in
-(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
-n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4
-(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H13: (drop n0 O x0
-e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
-b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
-e2))))))) (drop_drop (Bind b) n0 x0 e H13 t))) (\lambda (H13: (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0
-n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0
-(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
-T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x4))).(\lambda (H15:
-(drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H16: (subst0 (minus x1 (s x2 n0))
-v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
-(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
-(CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4)
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
-x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4))
-(drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H15 t) (eq_ind_r nat (S (s x2 n0))
-(\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H16 (s x2 (S
-n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
-n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
-k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x0
-(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
-C).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
-(drop n0 O x0 (CHead x4 x2 x5))).(\lambda (H16: (csubst0 (minus x1 (s x2 n0))
-v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
-(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
-(CHead x0 (Bind b) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
-b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
-x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H15 t) (eq_ind_r nat (S (s x2
-n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H16 (s
-x2 (S n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex4_5 K C C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2
-k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
-n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T
-T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
-b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
-e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
-(drop n0 O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2 n0))
-v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 n0)) v x3 x4)).(eq_ind_r C
-(CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t)
-c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0
-(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x0 (Bind b) t)
-(CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e0 k0 u)))))) (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
-(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
-(CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1
-e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) x2 x3 x4 x5 x6
-(refl_equal C (CHead x3 x2 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x6)
-H15 t) (eq_ind_r nat (S (s x2 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind
-b) x1) n1) v x5 x6)) H16 (s x2 (S n0)) (s_S x2 n0)) (eq_ind_r nat (S (s x2
-n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H17 (s
-x2 (S n0)) (s_S x2 n0)))) e H14)))))))))) H13)) H12)))))) (\lambda (f:
-F).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(\lambda (H10: ((\forall (c3:
-C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
-x1))).(let H12 \def (H10 x0 v H6 e H9) in (or4_ind (drop (S n0) O x0 e)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1
-(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2)))))))) (\lambda (H13: (drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O
-(CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x0 e H13
-t))) (\lambda (H13: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x1 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S n0))) v u
-w))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Flat f) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3:
-C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2
-x4))).(\lambda (H15: (drop (S n0) O x0 (CHead x3 x2 x5))).(\lambda (H16:
-(subst0 (minus x1 (s x2 (S n0))) v x4 x5)).(eq_ind_r C (CHead x3 x2 x4)
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
-k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
-f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 x2
-x4)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x3 x2 x4) (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (ex3_4_intro K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))
-x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead
-x3 x2 x5) H15 t) H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0
-(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
-x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4:
-C).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
-(drop (S n0) O x0 (CHead x4 x2 x5))).(\lambda (H16: (csubst0 (minus x1 (s x2
-(S n0))) v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop
-(S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
-(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
-(CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S
-n0) O (CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
-f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2
-x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H15 t) H16)) e H14))))))))
-H13)) (\lambda (H13: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
-(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0
-(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K
-C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
-f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15:
-(drop (S n0) O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2
-(S n0))) v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 (S n0))) v x3
-x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead
-x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O
-(CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5)
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1
-e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f)
-n0 x0 (CHead x4 x2 x6) H15 t) H16 H17)) e H14)))))))))) H13)) H12)))))) k
-(drop_gen_drop k c e t n0 H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3:
-(ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s
-k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
-c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
-nat).(\lambda (H4: (eq nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k
-x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c
-x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0
-(S n0))) v e1 e2))))))))) (let H8 \def (eq_ind nat i (\lambda (n1:
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S
-n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1
-(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H4) in (let H9 \def (eq_ind
-nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k
-x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
-(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0
-(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to
-(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2)
-(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2))))))
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0
-u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1
-e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1
-k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0)
-(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4
-K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u))))))
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
-k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w))))))
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda
-(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b)
-x2))).(let H13 \def (IHn x2 (le_S_n (S n0) x2 H12) c x1 v H7 e H10) in
-(or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
-n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4
-(drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x1
-e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
-b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
-e2))))))) (drop_drop (Bind b) n0 x1 e H14 x0))) (\lambda (H14: (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0
-n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1
-(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x5))).(\lambda (H16:
-(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H17: (subst0 (minus x2 (s x3 n0))
-v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
-(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O
-(CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5)
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x2) (s k0 (S n0))) v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5))
-(drop_drop (Bind b) n0 x1 (CHead x4 x3 x6) H16 x0) (eq_ind_r nat (S (s x3
-n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H17 (s
-x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
-n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
-k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1
-(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
-C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
-(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 n0))
-v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
-(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O
-(CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3
-x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
-b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3
-x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H16 x0) (eq_ind_r nat (S (s
-x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H17
-(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead
-e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0
-n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T
-T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
-b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
-e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
-T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
-(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 n0))
-v x6 x7)).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C
-(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0)
-c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0
-(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O
-(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
-(CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6)
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1
-e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
-x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3
-x7) H16 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s
-(Bind b) x2) n1) v x6 x7)) H17 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s
-x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18
-(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f:
-F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3:
-C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0:
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f)
-x2))).(let H13 \def (H11 x1 v H7 e H10) in (or4_ind (drop (S n0) O x1 e)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w)))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2
-(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e)
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2)))))))) (\lambda (H14: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O
-(CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Flat f) x2) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H14
-x0))) (\lambda (H14: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
-e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus x2 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
-w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
-C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3
-x5))).(\lambda (H16: (drop (S n0) O x1 (CHead x4 x3 x6))).(\lambda (H17:
-(subst0 (minus x2 (s x3 (S n0))) v x5 x6)).(eq_ind_r C (CHead x4 x3 x5)
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T
-T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
-k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
-f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
-x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
-(ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
-w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1
-(CHead x4 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C
-C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0
-(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead
-x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0
-(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5:
-C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16:
-(drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3
-(S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop
-(S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u))))))
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0
-(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O
-(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))))
-(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K
-C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
-(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1
-(CHead x5 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C
-C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u
-w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1
-e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4
-(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
-x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
-(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4:
-C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e
-(CHead x4 x3 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 x3
-x7))).(\lambda (H17: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda
-(H18: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3
-x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K
-C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w))))))
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
-k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
-f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3
-x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0:
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))))
-(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w))))))
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5
-x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3
-x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))) k (drop_gen_drop k c e
-t n0 H2) H8 H9) i H4))) c2 H5)))))))) H3)) (csubst0_gen_head k c c2 t v i
-H1))))))))))) c1)))))) n).
-
-theorem csubst0_drop_eq:
- \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
-n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
-C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1
-e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
-c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1
-e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
-c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
-(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0
-y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c:
-C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0
-t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0:
-K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2:
-T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4
-(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
-c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f)
-u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
-k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1)
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2)
-(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda
-(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
-u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat
-(S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with
-[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4
-(drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
-c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2)
-(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
-u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind
-b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b)
-u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i:
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
-i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind
-nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O
-(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f)
-u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0
-(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
-(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c
-(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f)
-u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat
-f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq
-C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
-(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f)
-u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1))
-(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k:
-K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
-C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
-i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall
-(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u)
-(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0
-i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0
-i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f)
-u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
-T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
-e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
-C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
-(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u)
-(CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0
-(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S
-i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i)
-(S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1
-(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1
-e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3:
-C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3
-c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i
-O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
-(drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0
-(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
-(w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0
-n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
-C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
-T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
-e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
-nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
-nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
-(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
-T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
-u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat
-f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
-u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3
-(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0:
-T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
-(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C
-T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
-(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u)
-(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C
-(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4
-(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
-u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0)
-u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u))
-(drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
-K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
-T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
-C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
-C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
-(s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0
-u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat
-f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f)
-u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s
-k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1)
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2)
-(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
-i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
-\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
-in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind
-b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i)
-(S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead
-c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2))))))))
-H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda
-(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3:
-C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq
-nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i
-c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T
-T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2
-(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
-e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda
-(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda
-(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u)))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u
-w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def
-(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8
-\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in
-(eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2)
-(CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0
-u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F
-C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3
-(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f)
-u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
-u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
-u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1))
-(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
-H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
-((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
-(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda
-(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall
-(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
-nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort
-n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1)
-e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop
-(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0)
-O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4
-(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in
-nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1))
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f)
-u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c:
-C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2)
-\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda
-(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v
-(CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t)
-e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s
-k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2))))
-(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda
-(_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda
-(_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
-n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq
-nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k
-u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
-nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat
-(S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5:
-(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S
-n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
-n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
-k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c
-e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat
-nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
-\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def
-(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H8) in
-(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 F C T T (\lambda
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c
-(Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H6 x0)))))))
-(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq
-nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0:
-nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda
-(n1: nat).(subst0 n1 v t x0)) H5 (S n0) H8) in (or4_intro0 (drop (S n0) O
-(CHead c (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) x0)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))) (drop_drop (Flat f) n0 c e H6 x0))))))) k (drop_gen_drop k c
-e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
-v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0)
-(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t))))
-(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O
-c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0)
-(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1
-v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0
-e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
-n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c
-e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat
-nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
-\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def
-(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H8) in (let
-H10 \def (IHn c x0 v H9 e H6) in (or4_ind (drop n0 O x0 e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11:
-(drop n0 O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x0 e H11
-t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
-x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x0 (Bind
-b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop n0 O
-x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 x5)).(eq_ind_r C
-(CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind
-b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
-x2) x4)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
-x3 (Flat x2) x4) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
-x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4))
-(drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) e H12))))))))
-H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
-x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0
-(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop n0 O
-x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 x4)).(eq_ind_r C
-(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind
-b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat
-x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
-x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t)
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2)
-x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
-(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H13 t) H14)) e H12))))))))
-H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
-O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
-(H13: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v x5
-x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5)
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
-(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
-x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
-(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
-x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0
-x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10)))))))
-(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq
-nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0:
-nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda
-(n1: nat).(csubst0 n1 v c x0)) H5 (S n0) H8) in (let H10 \def (H x0 v H9 e
-H6) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11:
-(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat
-f) n0 x0 e H11 t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
-(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
-O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda
-(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop
-(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4
-x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
-(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3
-(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14))
-e H12)))))))) H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f0: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
-C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
-(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
-(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop
-(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3
-x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3
-(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
-C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
-(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead
-x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H13 t)
-H14)) e H12)))))))) H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0
-(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
-f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda
-(H13: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v
-x5 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2)
-x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
-(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2)
-x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat
-f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
-f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat
-f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
-x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0
-x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10)))))))
-k (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
-k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
-c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
-(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
-C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F
-C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
-C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4:
-(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6:
-(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop
-(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S
-n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c
-e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(let H9 \def (f_equal nat
-nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
-\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H10 \def
-(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H9) in (let
-H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in
-(let H12 \def (IHn c x1 v H10 e H7) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T
-T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0)
-e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13:
-(drop n0 O x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H13
-x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
-x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind
-b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
-T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop n0 O
-x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 x6)).(eq_ind_r C
-(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
-b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
-x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
-O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
-x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
-x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5))
-(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) H16)) e H14))))))))
-H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
-x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1
-(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
-T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H15: (drop n0 O
-x1 (CHead x5 (Flat x3) x6))).(\lambda (H16: (csubst0 O v x4 x5)).(eq_ind_r C
-(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind
-b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat
-x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
-O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
-x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0)
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
-x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
-x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6))
-(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14))))))))
-H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
-O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
-O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
-T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda
-(H15: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H16: (subst0 O v x6
-x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6)
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T
-T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
-O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O
-(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3)
-x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
-(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
-x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0
-x1 (CHead x5 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13))
-H12)))))))) (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c
-e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(let H9 \def (f_equal nat
-nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H8) in (let H10 \def
-(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 (S n0) H9) in
-(let H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S
-n0) H9) in (let H12 \def (H x1 v H10 e H7) in (or4_ind (drop (S n0) O x1 e)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: (drop (S n0) O
-x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H13
-x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda
-(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0)
-O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda
-(x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop
-(S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5
-x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead
-x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0)
-u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4
-(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H15 x0)
-H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind
-F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0)
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4:
-C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat
-x3) x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda
-(H16: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0:
-C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda
-(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat
-f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0)
-u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
-x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1
-(CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) H13)) (\lambda (H13:
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0)
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda
-(x6: T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3)
-x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda
-(H16: (subst0 O v x6 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C
-(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat
-f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat
-f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0)
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat
-x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
-n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1
-(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C
-C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))))
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7
-(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5
-(Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) H12)))))))) k
-(drop_gen_drop k c e t n0 H1) H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2
-t v (S n0) H0))))))))))) c1)))) n).
-
-theorem csubst0_drop_eq_back:
- \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
-n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
-C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2
-e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O
-c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2
-e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
-(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O
-O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
-(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1
-u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda
-(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t:
-T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0
-t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda
-(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall
-(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s
-k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2))
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4:
-T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda
-(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
-u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3:
-T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2)
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0
-u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0
-u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
-T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i:
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
-i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def
-(eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in
-(False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b)
-u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead
-c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C
-C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
-(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b)
-u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq
-C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i)
-(CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
-u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f:
-F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i
-O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O
-H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f)
-u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0
-(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
-T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0)
-u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
-T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2
-(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead
-e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c
-(Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
-(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C
-C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
-(CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1)
-(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
-(CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c
-(Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f)
-u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0)
-u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
-T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2))
-(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k:
-K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4:
-C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop
-i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to
-(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead
-c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f)
-u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s
-k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0
-i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f)
-u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0
-e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3:
-C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
-(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u)
-(CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S
-i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f)
-u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
-T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S
-i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead
-e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u)
-(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
-T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i:
-nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2:
-(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i
-v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1
-(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1
-e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def
-(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0
-n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
-C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
-e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
-nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0:
-nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
-(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
-u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
-u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f)
-u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4
-(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O
-(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C
-C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C
-(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u)
-(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
-(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3
-(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f)
-u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0)
-u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u))
-(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k:
-K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1:
-T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4:
-C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
-C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
-(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
-(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
-u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0
-(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
-(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2)
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat
-f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0
-u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0
-u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
-T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i:
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0
-i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
-(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3:
-T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
-(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
-\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda
-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5)
-in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind
-b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i)
-(S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3
-u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i)
-(S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2
-(Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1
-(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i:
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
-i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3
-c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
-(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3:
-T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0)
-u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3:
-T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
-i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i
-O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4
-(drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0
-(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda
-(u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0
-c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4
-(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0)
-u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3:
-T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1
-e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0:
-nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda
-(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0:
-nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F
-C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq
-C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3
-(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1)
-(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0
-n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
-n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O
-(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
-u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0)
-u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
-T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2
-(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead
-e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f)
-u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1)
-(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2))
-(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2
-H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
-((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
-(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind
-(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to
-(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f)
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
-nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort
-n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort
-c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat
-f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S
-n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O
-c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e:
-C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S
-n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat
-(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
-nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
-(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4
-(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda
-(x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2
-(CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2
-(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind
-(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to
-(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b:
-B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r
-(Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match
-e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1)
-\Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1
-(\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0)
-O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead
-e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f:
-F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r
-(Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S
-n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v
-t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
-(Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2))
-(\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0)
-(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t))))
-(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat
-(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k
-t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0)
-(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1
-v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1
-(CHead x0 k t) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to
-((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
-C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
-e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b)
-x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat
-nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
-\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def
-(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let
-H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
-(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
-(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
-(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
-x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15:
-(subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4
-(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
-b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f)
-u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
-x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c
-(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq
-C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2)
-x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5)
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
-(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
-x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
-f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5
-(refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3
-(Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T
-T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
-T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O
-c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16:
-(csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0:
-C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
-b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f)
-u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
-x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0
-c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))))
-(\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8:
-(drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0:
-nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1:
-nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in
-(or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead
-e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0)
-O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e
-H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda
-(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
-(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
-(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
-x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda
-(H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0:
-C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat
-f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
-(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13))))))))
-H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0)
-O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
-(x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop
-(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3
-x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4
-(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0)
-u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0)
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
-x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c
-(CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
-(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4
-(Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2)
-x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3
-x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O
-(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4
-(Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0)
-u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0)
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6))
-(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e
-H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6))))))))
-H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda
-(j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0)
-O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda
-(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2
-(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2
-v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1
-(CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to
-((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
-C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
-e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b)
-x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat
-nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
-\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def
-(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let
-H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10)
-in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
-(Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
-(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
-(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3)
-x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17:
-(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4
-(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
-b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f)
-u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2))))))
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
-x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c
-(CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14:
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq
-C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3)
-x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6)
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
-(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
-x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
-f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6
-(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4
-(Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T
-T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
-e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
-T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O
-c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18:
-(csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0:
-C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
-b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f)
-u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
-x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0
-c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13))))))))
-(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9:
-(drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0:
-nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1:
-nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2
-(\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1
-v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
-(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
-(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
-e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
-(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
-(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda
-(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4
-(Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3)
-x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6)
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
-(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O
-(CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
-x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
-x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
-(_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat
-f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
-x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
-x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead
-x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t)
-H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
-C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
-(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
-(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
-(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3)
-x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda
-(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0:
-C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat
-f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop
-(Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14))
-(\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0)
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C
-C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
-n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
-(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
-T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S
-n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda
-(H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0:
-C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat
-f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
-x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat
-x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e
-H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7))))))))))
-H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n).
-
-theorem csubst0_drop_lt_back:
- \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O
-c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n)
-v e1 e2)) (\lambda (e1: C).(drop n O c1 e1))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
-\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
-(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
-c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
-c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda
-(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1
-c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0:
-nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2))
-(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C
-(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1)))
-(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O
-c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl
-c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt
-n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1
-c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C
-(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
-c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda
-(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v
-c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c
-e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1:
-C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda
-(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O
-(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2))
-(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c:
-C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
-(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C
-(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop
-(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
-C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda
-(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda
-(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0)
-O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1
-e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3:
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda
-(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
-T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead
-c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s
-k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t
-x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2
-(CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall
-(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S
-n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0
-(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))
-H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0)
-n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S
-n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v
-e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda
-(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to
-(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
-(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1:
-C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0
-n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0)
-O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3:
-C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3:
-C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
-C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
-(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda
-(H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind
-b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda
-(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2
-H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0:
-T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
-e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
-(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c
-e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop
-(r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2)
-(ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1:
-C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12
-t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda
-(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))
-(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
-v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead
-c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s
-k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c
-x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2
-(CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall
-(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S
-n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0
-(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))
-H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0)
-n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S
-n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v
-e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda
-(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to
-(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C
-(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1:
-C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0
-n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0)
-O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3:
-C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3:
-C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
-C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
-(S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda
-(H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1
-H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2
-C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0
-O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
-(Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O
-(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1
-e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop
-(Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0
-(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C
-(\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
-c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
-(Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x
-e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind
-b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda
-(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1:
-C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
-(Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14))
-H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0:
-T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
-e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
-(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c
-e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop
-(r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def
-H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus
-x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0)
-O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0))
-v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))))
-(\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat
-f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2))
-(\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat
-f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1
-(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C
-(\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop
-(S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda
-(e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
-(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1
-(S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0)
-O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0))
-v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))
-(ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda
-(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0
-c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i
-H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0)
-O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1
-e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0:
-T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k
-x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
-x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda
-(c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind
-nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c
-c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3)
-(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1:
-C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat
-i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2)
-(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
-(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall
-(v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3
-e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s
-k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to
-((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O
-(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0))
-v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda
-(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b)
-x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0)
-O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1
-e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S
-n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x
-\def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in
-(or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1
-e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b)
-t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1:
-C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c
-e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c
-(Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C
-(\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O
-c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2))
-(\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t)
-e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1:
-C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16:
-(csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror
-(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0
-(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t)
-e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda
-(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0
-c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall
-(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3:
-C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1:
-C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop
-(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda
-(H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in
-(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c
-e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
-(CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl
-(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0
-(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f)
-t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda
-(e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
-c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2))
-(\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f)
-t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda
-(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda
-(H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c
-x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O
-(CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2
-(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x
-H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10
-(drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c
-c2 t v i H1))))))))))) c1)))))) n).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst0/defs.ma".
-
-theorem csubst0_gen_sort:
- \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0
-i v (CSort n) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n)
-(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y:
-C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda
-(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P)))))
-(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda
-(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq
-C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in
-(False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1
-c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda
-(H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
-(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1
-u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1
-c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead
-c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P
-H5))))))))))))) i v y x H0))) H)))))).
-
-theorem csubst0_gen_head:
- \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
-(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
-nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
-nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
-u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))))
-(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
-u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
-c2))))))))))))
-\def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1)
-x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda
-(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda
-(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_:
-nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j
-v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_:
-nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
-(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j:
-nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y
-x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda
-(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k
-j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda
-(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2:
-T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_:
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda
-(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2:
-T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C
-(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c |
-(CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let
-H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0
-u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match
-e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
-t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq
-K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))
-(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3))))
-(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2:
-C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2:
-C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda
-(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
-u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1
-c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1
-u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda
-(_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda
-(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C
-(CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j:
-nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda
-(c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda
-(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_:
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
-(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3))))
-(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2:
-C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2:
-C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
-(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
-u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1
-c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0)
-(s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1
-k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0
-(refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c
-H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0:
-C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0
-c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda
-(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_:
-nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
-v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda
-(u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u)
-(CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
-C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _)
-\Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u)
-(CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0
-c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_:
-T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda
-(_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda
-(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C
-(CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda
-(c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def
-(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat
-(\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda
-(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
-c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3
-T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
-k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
-c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0
-i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat
-(\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2:
-T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2:
-T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda
-(_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda
-(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda
-(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
-(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2))))
-(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3:
-C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
-(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
-c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0)
-(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3
-k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0
-(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u
-H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0:
-T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0
-u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0
-c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda
-(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_:
-nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j
-v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda
-(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda
-(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda
-(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k
-u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _)
-\Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0)
-(CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0
-c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to
-(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j))))
-(\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3:
-T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
-j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda
-(j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0
-(\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0
-(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1:
-K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k
-j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k
-u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3:
-C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda
-(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
-c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat
-(s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2)
-(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))))
-(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
-(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1))))
-(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat
-(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k
-j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k
-u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j:
-nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
-nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda
-(_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1
-c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12
-H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))).
-
-theorem csubst0_gen_S_bind_2:
- \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall
-(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to
-(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x
-(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2))
-(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
-C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_:
-T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1
-(Bind b) v1))))))))))))
-\def
- \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b)
-v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x
-c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda
-(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i
-v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1:
-C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1:
-T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0:
-(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x
-y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T
-(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind
-b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C
-x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
-T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1))))))))
-(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda
-(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i))
-\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1:
-T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1))))
-(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead
-c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1
-v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1:
-C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k:
-K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat
-(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b)
-v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2
-(Bind b) v2) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
-\Rightarrow t])) (CHead c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8:
-(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0:
-C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
-(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i
-v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2))))
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
-(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1:
-T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T
-u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K
-k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K
-(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2))
-(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C
-(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0
-u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
-T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1
-(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e
-in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n)
-\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda
-(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1:
-T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead
-c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda
-(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1:
-C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1:
-T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T
-(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind
-b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b)
-u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0:
-nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2:
-(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead
-c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2))
-(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
-C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2))))
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
-T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq
-nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b)
-v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
-(CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2
-(Bind b) v2) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9:
-(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t:
-T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
-(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i
-v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2))))
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
-T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C
-c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2))
-\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C
-c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
-(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
-C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
-T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead
-c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c:
-C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda
-(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b)
-(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda
-(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
-C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3
-(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1
-v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3:
-C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let
-H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda
-(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i)
-H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to
-((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i
-v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C
-(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3
-(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1
-v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3:
-C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in
-(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i
-H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda
-(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C
-(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind
-b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
-T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3
-c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead
-c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
-(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16
-(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7))
-H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda
-(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1:
-C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq
-nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda
-(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b)
-v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C
-c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1:
-T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3
-c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b)
-v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C
-(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0
-| (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6)
-in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u2) (CHead c2
-(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0
-c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C
-c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1
-v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3:
-C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2))))
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
-T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def
-(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14
-\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15
-\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10)
-in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0
-i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b)
-v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C
-(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
-C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
-T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1
-k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda
-(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0
-| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0
-(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to
-(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1
-(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2))
-(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_:
-C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_:
-T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead
-c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda
-(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0
-(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T
-(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind
-b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3
-c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2))))
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
-T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3:
-C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1:
-T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18
-(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8))
-H7)))))))))))))) y0 v x y H1))) H0))) H))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst0/clear.ma".
-
-include "LambdaDelta-1/csubst0/drop.ma".
-
-include "LambdaDelta-1/getl/fwd.ma".
-
-theorem csubst0_getl_ge:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1
-e) \to (getl n c2 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
-c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all
-c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0:
-C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1
-x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5:
-(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4))
-(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0:
-nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0:
-nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e))
-(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x)
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9:
-(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
-w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat
-x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_:
-(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
-(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3)
-H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9))
-(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u))))))
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0:
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x
-(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0)
-x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda
-(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e
-(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e
-(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F
-C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O
-c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f)
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0:
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O
-c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13:
-(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e))
-H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4)
-H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e
-x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n
-i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))).
-
-theorem csubst0_getl_lt:
- \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1
-e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
-c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all
-c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0:
-C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x:
-C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def
-(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K
-C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k:
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O
-c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B
-C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
-e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e)
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6:
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u
-w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda
-(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k
-n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0:
-K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x
-(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda
-(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda
-(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop
-n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear
-(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b:
-B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12:
-(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1
-(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4
-(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
-c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2
-(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0
-(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
-(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead
-e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2
-(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12))
-e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n
-O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v
-x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n
-c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1
-(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3)))))))
-x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k:
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind
-K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1
-e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0:
-K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x
-(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda
-(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda
-(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop
-n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear
-(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b:
-B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12:
-(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1
-(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4
-(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda
-(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
-c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2
-(CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0
-(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
-(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
-(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead
-e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n
-c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3))
-H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11:
-(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s
-(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let
-H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12
-(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1
-x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear
-x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4
-B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
-C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e
-H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2
-(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
-(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n))
-v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4:
-B).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H17: (eq C e
-(CHead x5 (Bind x4) x6))).(\lambda (H18: (clear x2 (CHead x5 (Bind x4)
-x7))).(\lambda (H19: (subst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x5
-(Bind x4) x6) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1
-(getl n c2 (CHead x5 (Bind x4) x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
-x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b)
-u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5
-x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind
-x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18
-f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda
-(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18:
-(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v
-x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2
-c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4)
-x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4)
-x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1
-(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))
-(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7
-(refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4)
-x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f
-x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda
-(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq
-C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4)
-x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20:
-(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7)
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
-(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
-x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
-x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7))
-(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11
-(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17))))))))))
-H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda
-(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
-n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k
-n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n
-c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda
-(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq
-C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda
-(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s
-x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
-(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4))
-\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1
-x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2
-(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3
-x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda
-(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3)
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u))))))
-(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
-(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u)))))))
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
-(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead
-e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
-(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda
-(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C
-T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))))
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4
-(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4)
-(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind
-b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2
-(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3
-x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda
-(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n)
-(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy
-i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e
-(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
-(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n))
-v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18:
-(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e
-(CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18:
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
-T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-(minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2
-(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e)
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda
-(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5)
-x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21:
-(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7)
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1
-(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
-x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b)
-u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6
-x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind
-x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20
-f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda
-(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20:
-(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v
-x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2
-c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5)
-x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
-(_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5)
-x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1
-(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))
-(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8
-(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5)
-x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f
-x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda
-(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq
-C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5)
-x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22:
-(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8)
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
-(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
-C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
-x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
-x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
-v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8))
-(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12
-(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19))))))))))
-H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
-
-theorem csubst0_getl_ge_back:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2
-e) \to (getl n c1 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
-c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all
-c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0:
-C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2
-x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5:
-(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3)
-H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0:
-nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0:
-nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e))
-(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O
-c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1)))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1
-e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9:
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
-T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1))))))
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
-u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f:
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat
-x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_:
-(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
-(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2)
-H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9))
-(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u))))))
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0:
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x
-(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0)
-x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda
-(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e
-(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v
-H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9:
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f:
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i
-O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0:
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O
-c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13:
-(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e))
-H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3)
-H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0
-x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n
-i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))).
-
-theorem csubst0_getl_lt_back:
- \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2
-e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1
-e2)) (\lambda (e1: C).(getl n c1 e1))))))))))))
-\def
- \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
-c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def
-(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e))
-(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda
-(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x
-\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in
-(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x))
-(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda
-(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))
-(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0
-(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda
-(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or
-(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2))
-(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0
-(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def
-(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in
-(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2))
-(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda
-(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1:
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))
-(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1:
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind
-C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0
-e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1
-e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11:
-(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl
-n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1:
-C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1
-e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8
-H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst0/defs.ma".
-
-theorem csubst0_snd_bind:
- \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
-(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c
-(Bind b) u1) (CHead c (Bind b) u2))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
-(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind
-b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b)
-u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S
-i))))))))).
-
-theorem csubst0_fst_bind:
- \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall
-(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1
-(Bind b) u) (CHead c2 (Bind b) u))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind
-b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b)
-u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))).
-
-theorem csubst0_both_bind:
- \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
-(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i
-v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b)
-u2))))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
-(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
-C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n:
-nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))
-(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S
-i))))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst0/defs.ma".
-
-inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def
-| csubst1_refl: csubst1 i v c1 c1
-| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst1/defs.ma".
-
-include "LambdaDelta-1/csubst0/fwd.ma".
-
-include "LambdaDelta-1/subst1/props.ma".
-
-theorem csubst1_gen_head:
- \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
-(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T
-C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
-C).(csubst1 i v c1 c2))))))))))
-\def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1)
-x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C
-(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
-C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2:
-C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_:
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1
-c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl
-i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1)
-c2)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i)
-(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2))))
-(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3:
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda
-(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C
-(\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
-C).(csubst1 i v c1 c3)))) (\lambda (H1: (ex3_2 T nat (\lambda (_: T).(\lambda
-(j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C
-c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1
-u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s
-k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2))))
-(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda
-(u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
-C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H2:
-(eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead c1 k x0))).(\lambda
-(H4: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2
-T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda
-(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
-C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 (\lambda (n:
-nat).(subst0 n v u1 x0)) H4 i (s_inj k i x1 H2)) in (ex3_2_intro T C (\lambda
-(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda
-(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
-C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single
-i v u1 x0 H5) (csubst1_refl i v c1))) c2 H3)))))) H1)) (\lambda (H1: (ex3_2 C
-nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
-j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3
-k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1:
-nat).(\lambda (H2: (eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead
-x0 k u1))).(\lambda (H4: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1)
-(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead
-c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda
-(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1
-(\lambda (n: nat).(csubst0 n v c1 x0)) H4 i (s_inj k i x1 H2)) in
-(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1)
-(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
-(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C
-(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H5))) c2
-H3)))))) H1)) (\lambda (H1: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
-k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
-c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k
-u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1:
-C).(\lambda (x2: nat).(\lambda (H2: (eq nat (s k i) (s k x2))).(\lambda (H3:
-(eq C c2 (CHead x1 k x0))).(\lambda (H4: (subst0 x2 v u1 x0)).(\lambda (H5:
-(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C
-(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
-C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x2 (\lambda (n:
-nat).(csubst0 n v c1 x1)) H5 i (s_inj k i x2 H2)) in (let H7 \def (eq_ind_r
-nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H4 i (s_inj k i x2 H2)) in
-(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0)
-(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
-(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C
-(CHead x1 k x0)) (subst1_single i v u1 x0 H7) (csubst1_sing i v c1 x1 H6))))
-c2 H3)))))))) H1)) (csubst0_gen_head k c1 c2 u1 v (s k i) H0)))) x H))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst1/props.ma".
-
-include "LambdaDelta-1/csubst0/getl.ma".
-
-include "LambdaDelta-1/subst1/props.ma".
-
-include "LambdaDelta-1/drop/props.ma".
-
-theorem csubst1_getl_ge:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1
-e) \to (getl n c2 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to
-(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
-(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
-(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))).
-
-theorem csubst1_getl_lt:
- \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1
-e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2:
-C).(getl n c2 e2)))))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to
-(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl
-n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S
-(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1
-e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2:
-C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1
-(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H))))
-(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda
-(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0:
-nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n
-c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind
-(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u:
-T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n
-c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2:
-C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)))
-(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S
-(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl
-(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
-T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
-T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S
-(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
-B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1
-(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0)
-x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1
-(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
-n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
-C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2:
-C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
-v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus
-i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1
-(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3:
-C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n))
-v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda
-(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1
-(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1
-(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0)
-x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1
-(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
-n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
-C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2:
-C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
-v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus
-i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T
-T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda
-(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2
-(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda
-(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
-(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
-(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2:
-C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind
-x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7:
-(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1
-x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2:
-C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2))))
-(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind
-x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4)
-(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind
-x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1
-H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))).
-
-theorem csubst1_getl_ge_back:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2
-e) \to (getl n c1 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to
-(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
-(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
-(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))).
-
-theorem getl_csubst1:
- \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c
-(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_:
-C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0
-a))))))))
-\def
- \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e:
-C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C
-(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0:
-C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind
-(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind
-Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0
-a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda
-(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n)
-(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2
-C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda
-(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda
-(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to
-(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda
-(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind
-(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O
-(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0:
-C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0:
-C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t:
-T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b)
-t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
-(CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b)
-t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind
-b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B)
-with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e
-(Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind
-Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in
-((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e
-(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u)
-H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t
-(\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0
-(CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0
-a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda
-(_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda
-(a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_:
-C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a:
-C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead
-c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u
-H3)))) H2)) H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e:
-C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind
-Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T
-(\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0:
-C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0:
-C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2:
-(subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead
-e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr)
-u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in
-(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0)))
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda
-(a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda
-(a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1:
-C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0
-x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0
-(Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))
-(CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O
-x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3))))
-H1)))))))) k)))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall
-(e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C
-(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0:
-C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind
-(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e
-(Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S
-n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))
-(\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n)
-(CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind
-Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u
-(CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0
-a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u:
-T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0:
-C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a:
-C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0:
-K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0
-t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_:
-C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a:
-C).(drop (S O) (S n) a0 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
-(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead
-e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in
-(ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C
-(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0)))
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
-T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u
-(getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
-(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0:
-C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda
-(_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda
-(a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda
-(H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C
-C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t)
-a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0
-(Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift
-(S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4))))
-H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
-T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr)
-u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T
-(\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C
-(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0)))
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
-T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e
-u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
-(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0:
-C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0:
-C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0:
-C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1:
-C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0
-x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u
-(CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S
-n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x)
-(csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat
-(S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst1/defs.ma".
-
-include "LambdaDelta-1/subst1/defs.ma".
-
-theorem csubst1_head:
- \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
-(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
-v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2))))))))))
-\def
- \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
-(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t:
-T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k
-i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2:
-C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c:
-C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i)
-v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1
-c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i
-c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
-t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1)
-(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2)
-(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v
-c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both
-k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))).
-
-theorem csubst1_bind:
- \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
-(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
-v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b)
-u2))))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
-(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
-C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n:
-nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))
-(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S
-i))))))))))).
-
-theorem csubst1_flat:
- \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
-(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
-v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f)
-u2))))))))))
-\def
- \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
-(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
-C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n:
-nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2)))
-(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/defs.ma".
-
-include "LambdaDelta-1/clear/fwd.ma".
-
-theorem csubt_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2))
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
-c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c
-e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2))
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3
-e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
-(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u)
-e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear
-(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind
-b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2))))
-(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda
-(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g
-c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))))
-(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def
-(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g
-e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1
-e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x:
-C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C
-(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f)
-u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
-C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall
-(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda
-(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3:
-(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1)
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2:
-C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt
-g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b)
-u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2)
-(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3))))))))))))
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_:
-((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2))
-(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda
-(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda
-(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst)
-t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2:
-C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2:
-C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4
-(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2
-H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1
-c2 H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/arity.ma".
-
-theorem csubt_csuba:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba
-g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
-c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda
-(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
-(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda
-(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b:
-B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
-T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u:
-T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u
-t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A
-(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g
-a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda
-(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g
-x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1)))))
-H4))))))))))) c1 c2 H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/defs.ma".
-
-inductive csubt (g: G): C \to (C \to Prop) \def
-| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n))
-| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall
-(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u))))))
-| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall
-(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g
-(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/fwd.ma".
-
-include "LambdaDelta-1/drop/fwd.ma".
-
-theorem csubt_drop_flat:
- \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall
-(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1
-(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
-(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
-\def
- \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0:
-nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1:
-C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f)
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
-c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1
-(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H
-(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let
-H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C
-(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g
-d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O
-c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x
-(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u)
-(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f)
-u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda
-(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2)
-\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u))
-\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2
-(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda
-(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall
-(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead
-d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u:
-T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind
-(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1)
-(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort
-n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5
-\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda
-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
-in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop
-(S n0) O (CSort n1) (CHead d2 (Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0)
-O (CHead d1 (Flat f) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda
-(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop
-(S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
-u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall
-(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f)
-u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S
-n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda
-(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead
-c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g
-d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
-(Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g
-d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
-(Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x
-(Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1
-(Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1:
-C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u)
-(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0)
-u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1
-x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
-(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead
-x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1
-(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3:
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u:
-T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
-u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S
-n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f)
-u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O
-(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5:
-(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2
-C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
-(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x
-(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead
-d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3:
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u:
-T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
-u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u
-t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda
-(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f)
-u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0
-O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f)
-u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O
-c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f)
-u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0
-c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0
-H5)))))))))))))) c1 c2 H0)))))) n))).
-
-theorem csubt_drop_abbr:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g
-c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind
-Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop
-n O c2 (CHead d2 (Bind Abbr) u)))))))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1:
-C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u:
-T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr)
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1
-c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1
-(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H
-(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in
-(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq
-C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2
-(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr)
-u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u)
-(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead
-d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3))))
-H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2:
-C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1
-(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda
-(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda
-(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c
-(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda
-(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O
-(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr)
-u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr)
-u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3:
-(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0)
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1)
-(CHead d2 (Bind Abbr) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1
-(Bind Abbr) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1:
-(csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S
-n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr)
-u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall
-(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind
-Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda
-(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop
-(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2
-(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda
-(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x
-(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
-(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4
-(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1
-u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3))))))))
-(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda
-(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr)
-u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S
-n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr)
-u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S
-n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind
-Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u)))))
-(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0
-H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g
-c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0
-(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda
-(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0
-(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
-(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt
-g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
-(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead
-x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0
-(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda
-(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1:
-C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead
-d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g
-c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0:
-T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind
-Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2
-(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda
-(H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
-(CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind
-Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1
-(Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
-
-theorem csubt_drop_abst:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g
-c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind
-Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n
-O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1:
-C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t:
-T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst)
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
-(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
-g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g
-c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1
-(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H
-(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in
-(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2:
-C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)))
-(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr)
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_:
-C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3
-g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O
-O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C
-(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt
-g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t)))
-(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
-(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda
-(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead
-d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr)
-u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2:
-C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead
-d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t)))
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead
-d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3))
-(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2)))
-(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda
-(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2:
-T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t)))
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t))))
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
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-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O
-O c (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C
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-(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0
-(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
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-(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
-(d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
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-x1)) H6 H7)) c2 H4))))))) H3)) H2))))))))) (\lambda (n0: nat).(\lambda (H:
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-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr)
-u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2:
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-(S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda
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-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
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-H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0
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-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
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-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
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-(Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1:
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-(u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0
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-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
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-Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2:
-C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2
-(Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7
-u2)))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex4_2_ind C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop
-n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1
-u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b)
-u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda
-(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1
-t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2)
-(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t)))
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda
-(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
-T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
-g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7
-u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead
-d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3:
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t:
-T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst)
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
-(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_:
-(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0:
-T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind
-Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
-n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g
-d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
-(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
-(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_:
-C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3
-g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2
-(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))))
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7:
-(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst)
-t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))))
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
-(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind
-Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda
-(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3
-(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0
-t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda
-(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0
-t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S
-n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))))
-(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda
-(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7:
-(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr)
-x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1
-t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))))
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
-(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_:
-C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3
-g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1)
-H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0
-(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/defs.ma".
-
-theorem csubt_gen_abbr:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g
-(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr)
-v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2:
-C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))
-(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c:
-C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
-(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
-e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind
-Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2:
-C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
-(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2
-C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
-C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C
-(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1
-| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3)
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind
-Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1
-e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k
-t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K
-(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v)
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def
-(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C
-(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt
-g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g
-c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr)
-v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3
-(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5))
-H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1
-c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
-(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
-e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1
-(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
-\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr)
-v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2)
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)))))))))))
-(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_:
-(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u
-t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr)
-v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match
-ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
-_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C
-(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v)))
-(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))).
-
-theorem csubt_gen_abst:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g
-(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead
-e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda
-(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
-e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda
-(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind
-Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda
-(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1
-e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_:
-C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3
-g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or
-(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
-C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1:
-(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst)
-v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2
-(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda
-(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
-T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))
-H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1
-c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C
-(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
-g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
-(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
-(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
-(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1
-(Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
-C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
-\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
-(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
-(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2:
-C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g
-e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda
-(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2))))
-(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda
-(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
-v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind
-Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
-(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let
-H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst)
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda
-(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
-T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
-(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3
-(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
-C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1
-(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2
-(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
-e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
-v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1
-(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
-\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst)
-v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b)
-u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T
-(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
-(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
-(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3:
-C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind
-Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
-(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u:
-T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u
-t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst)
-v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
-(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind
-Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9
-\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def
-(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def
-(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def
-(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2
-C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
-C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
-C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind
-C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda
-(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda
-(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
-e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
-(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind
-Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt
-g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
-C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind
-Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))).
-
-theorem csubt_gen_flat:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall
-(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C
-c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead
-e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C
-(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
-e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda
-(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda
-(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1
-e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f)
-v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2:
-C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
-(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C
-(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
-e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k
-u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _)
-\Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u)
-(CHead e1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in
-(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v
-(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat
-f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda
-(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v)))
-(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c:
-C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
-(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in
-(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
-(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f)
-v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f)
-v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3:
-C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f)
-v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda
-(e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind
-Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind
-Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2:
-C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2:
-C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda
-(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2
-C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
-e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u
-t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t)
-(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2:
-C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2:
-C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))).
-
-theorem csubt_gen_bind:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
-(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
-(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_:
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y
-c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
-C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
-v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
-(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
-(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
-T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
-(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
-in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
-(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
-b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let
-H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
-(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
-C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b:
-B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1)
-v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
-(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1
-(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void
-b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c:
-C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
-H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9)
-in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind
-b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b)
-u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2))
-H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
-(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3
-B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
-e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u
-t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst)
-t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match
-e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _
-_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
-((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
-C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
-in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
-\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
-((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead
-c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst
-b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0:
-T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0:
-T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c:
-C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c:
-C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
-H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10)
-in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b)
-v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
-C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
-(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
-e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15))))))))))
-H7)) H6))))))))))) y c2 H0))) H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/clear.ma".
-
-include "LambdaDelta-1/csubt/drop.ma".
-
-include "LambdaDelta-1/getl/clear.ma".
-
-theorem csubt_getl_abbr:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
-(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g
-c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n
-c2 (CHead d2 (Bind Abbr) u)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
-(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e:
-C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u)))
-(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x:
-C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
-Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1
-(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
-u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda
-(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1
-(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
-u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
-Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1
-(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr)
-u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear
-(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1
-c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2
-(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1
-(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1
-(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
-\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
-Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u)
-t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda
-(c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda
-(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
-(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr
-H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c
-(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))
-(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2
-(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n
-c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1
-u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7)))))
-(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda
-(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5
-in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to
-(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda
-(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall
-(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1:
-C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
-C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c:
-C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat
-f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u)
-(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def
-(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u)
-H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2))
-(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2:
-C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13:
-(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C
-(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt
-g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2
-(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def
-(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15)
-in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2
-(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u)
-c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda
-(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall
-(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1:
-C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
-C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0
-(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e:
-C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_:
-B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t)))))
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
-(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4:
-T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0
-O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10
-(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead
-x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr)
-u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4)
-x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5
-x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
-(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7
-(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5
-(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21
-\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr)
-u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl
-n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22
-(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23)))))
-H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1
-H2)))) H0))))))).
-
-theorem csubt_getl_abst:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall
-(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g
-c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda
-(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e:
-C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t)))
-(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))
-(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead
-d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c
-(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2
-C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1
-(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst)
-t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0:
-C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t))
-\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
-t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1
-(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst)
-t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear
-(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1
-c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n
-c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5:
-(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b)
-t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
-(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b)
-t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0)
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind
-Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t)
-t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1
-x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r
-T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let
-H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t)))
-H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1
-(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
-d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2
-C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2
-(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
-(\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O c2
-(CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))
-(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2
-(CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t)
-(CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda
-(H16: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
-(d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
-d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
-(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O
-c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20:
-(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2
-H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18
-(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12
-d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1
-(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead
-d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n
-O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2
-C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1:
-C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1
-c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl
-n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda
-(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9:
-(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2))
-H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in
-(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0
-(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g
-(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C
-(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2:
-C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2:
-C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13:
-(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C
-(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt
-g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2)))
-(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda
-(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda
-(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
-T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
-(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t)))
-(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2
-(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
-Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
-(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3
-(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2
-(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead
-d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O
-c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15:
-(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr)
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_:
-C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3
-g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2:
-C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind
-Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4
-t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c:
-C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
-Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
-(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_:
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3
-(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14)))))
-H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1
-(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr)
-u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2:
-C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9:
-(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10:
-(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9)
-in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1
-(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
-T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2
-C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
-d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12:
-(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0
-(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind
-x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4)
-e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda
-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
-(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4)
-x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5
-x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda
-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
-(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5
-(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def
-(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18)
-in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
-(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead
-d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t)))
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2
-(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt
-g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2
-C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
-d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24:
-(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2:
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst)
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
-(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
-g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
-C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2
-x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C
-T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
-C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
-d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
-(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda
-(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
-(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
-(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g
-d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9:
-C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0
-x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda
-(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2))
-(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda
-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))
-(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
-(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3
-g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr)
-x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14)))))))
-H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/getl.ma".
-
-include "LambdaDelta-1/pc3/left.ma".
-
-theorem csubt_pr2:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
-t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2:
-C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
-C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0
-c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl
-i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_:
-(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta
-c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
-
-theorem csubt_pc3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
-t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0:
-T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t:
-T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4:
-T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1
-c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1
-c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4
-(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1
-t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall
-(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda
-(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0
-t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/defs.ma".
-
-theorem csubt_refl:
- \forall (g: G).(\forall (c: C).(csubt g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0))
-(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0
-c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/pc3.ma".
-
-include "LambdaDelta-1/csubt/props.ma".
-
-theorem csubt_ty3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
-t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda
-(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda
-(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u:
-T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2:
-C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3
-t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1
-c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda
-(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c
-c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda
-(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g
-d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c
-c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda
-(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
-u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5:
-(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g
-n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
-Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2:
-((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2:
-C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0
-c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2:
-C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n
-c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0
-u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n)
-(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2))
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda
-(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst)
-u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6:
-(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g
-n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n
-c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0
-u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T
-(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda
-(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
-T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3
-g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr)
-x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g
-n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2:
-C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda
-(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0
-t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1
-c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b)
-u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_:
-(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2
-w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind
-Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v
-(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c
-c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
-C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda
-(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4:
-T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c
-c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c
-c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
-
-theorem csubt_ty3_ld:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u
-v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1
-t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H:
-(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead
-c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead
-c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubv/defs.ma".
-
-include "LambdaDelta-1/clear/fwd.ma".
-
-theorem csubv_clear_conf:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1:
-B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to
-(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
-d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2
-(Bind b2) v2))))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall
-(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2:
-B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2)
-v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda
-(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1)
-v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2:
-B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2)
-v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3
-c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear
-c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2:
-C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
-C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda
-(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0:
-T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1)
-v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c]))
-(CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3
-(CHead d1 (Bind b1) v0) v1 H2)) in ((let H4 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b1 |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
-b) \Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0)
-(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1
-H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow
-t])) (CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void
-c3 (CHead d1 (Bind b1) v0) v1 H2)) in (\lambda (_: (eq B b1 Void)).(\lambda
-(H7: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda (b2: B).(\lambda
-(d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind b2)
-v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_:
-T).(csubv c3 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear
-(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))) Void c4 v2 H0
-(clear_bind Void c4 v2)) d1 H7)))) H4)) H3)))))))))))) (\lambda (c3:
-C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (_: ((\forall (b1:
-B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind b1) v1)) \to
-(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
-d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2
-(Bind b2) v2)))))))))))).(\lambda (b1: B).(\lambda (_: (not (eq B b1
-Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b0:
-B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear (CHead c3 (Bind b1)
-v1) (CHead d1 (Bind b0) v0))).(let H4 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
-(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1)
-v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in ((let H5 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-b0])])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3
-(CHead d1 (Bind b0) v0) v1 H3)) in ((let H6 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 |
-(CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1)
-v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in (\lambda (_: (eq
-B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B
-C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda
-(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2)
-(CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda
-(d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda (b3: B).(\lambda (d2:
-C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind b3)
-v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) H4)))))))))))))))
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (H1:
-((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1
-(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear
-c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda (f1: F).(\lambda (f2:
-F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1:
-C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1
-(Bind b1) v0))).(let H_x \def (H1 b1 d1 v0 (clear_gen_flat f1 c3 (CHead d1
-(Bind b1) v0) v1 H2)) in (let H3 \def H_x in (ex2_3_ind B C T (\lambda (_:
-B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2:
-B).(\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 (Bind b2) v3)))))
-(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
-d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4
-(Flat f2) v2) (CHead d2 (Bind b2) v3)))))) (\lambda (x0: B).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H4: (csubv d1 x1)).(\lambda (H5: (clear c4
-(CHead x1 (Bind x0) x2))).(ex2_3_intro B C T (\lambda (_: B).(\lambda (d2:
-C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
-C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2)
-v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind x0) x2) H5 f2 v2)))))))
-H3))))))))))))))) c1 c2 H))).
-
-theorem csubv_clear_conf_void:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1:
-C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda
-(v2: T).(clear c2 (CHead d2 (Bind Void) v2))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c
-(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2
-(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1:
-T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort
-(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead
-d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0:
-(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3
-(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2
-(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1:
-C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1
-(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
-\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind Void) v1)
-(clear_gen_bind Void c3 (CHead d1 (Bind Void) v0) v1 H2)) in ((let H4 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind
-Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind
-Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c:
-C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2:
-C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void)
-v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2)))
-(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2
-(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3)))))))))))
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_:
-((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to
-(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2:
-C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda
-(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear
-(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Void) v0)
-(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1
-H3)) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Void | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead d1 (Bind Void) v0)
-(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1
-H3)) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow
-t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3
-(CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq B Void b1)).(\lambda
-(H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda (v3: T).(clear
-(CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 \def (eq_ind_r
-B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let H10 \def (match
-(H9 (refl_equal B Void)) in False return (\lambda (_: False).(ex2_2 C T
-(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) (\lambda (d2: C).(\lambda
-(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) with
-[]) in H10)) d1 H8)))) H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1:
-T).((clear c3 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2:
-C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear
-c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda (f1: F).(\lambda (f2:
-F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0:
-T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind Void)
-v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind Void) v0)
-v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2
-(Bind Void) v3)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1
-d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead
-d2 (Bind Void) v3))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (csubv
-d1 x0)).(\lambda (H5: (clear c4 (CHead x0 (Bind Void) x1))).(ex2_2_intro C T
-(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda
-(v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4
-(clear_flat c4 (CHead x0 (Bind Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1
-c2 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-inductive csubv: C \to (C \to Prop) \def
-| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n))
-| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall
-(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind
-Void) v2))))))
-| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall
-(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall
-(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2)))))))))
-| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall
-(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1
-(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubv/props.ma".
-
-include "LambdaDelta-1/drop/fwd.ma".
-
-theorem csubv_drop_conf:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1:
-C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1
-e2)) (\lambda (e2: C).(drop h O c2 e2))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h
-O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O
-c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda
-(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq
-nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O
-(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h
-O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C
-(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2))))
-(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2))
-(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2:
-C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n)
-(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort
-n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3
-c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to
-(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4
-e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h:
-nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda
-(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2:
-C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2)
-e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C
-(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c
-e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C
-(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop
-O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same
-c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl
-(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0
-O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2))
-(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3:
-(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind
-Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in
-(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4
-e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O
-(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1
-x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1
-e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5
-(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda
-(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall
-(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2:
-C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1:
-B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h
-O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead
-c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2:
-C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead
-c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2
-C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind
-b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1)
-e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind
-b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2)
-v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0:
-nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C
-(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind
-b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1)
-e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1
-h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2))
-(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2))
-(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x:
-C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C
-(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4
-(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h
-H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3
-c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to
-(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4
-e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat
-f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1)
-e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O
-(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1)
-v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2)
-e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2))
-(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2)
-v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1
-(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda
-(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2:
-C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2)
-e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x
-\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in
-(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2:
-C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda
-(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda
-(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda
-(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2)
-v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h
-H2)))))))))))) c1 c2 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubv/clear.ma".
-
-include "LambdaDelta-1/csubv/drop.ma".
-
-include "LambdaDelta-1/getl/fwd.ma".
-
-theorem csubv_getl_conf:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1:
-B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1
-(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl
-i c2 (CHead d2 (Bind b2) v2)))))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1:
-B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i
-c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind
-b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e:
-C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda
-(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
-C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x:
-C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1)
-v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in
-(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2))
-(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1
-d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead
-d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda
-(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3)
-in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2:
-C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2:
-C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda
-(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2)
-v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8:
-(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro
-B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))))
-(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind
-b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6
-H9))))))) H7)))))) H4)))))) H1))))))))).
-
-theorem csubv_getl_conf_void:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1:
-C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1))
-\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2:
-C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1:
-C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1
-(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i
-H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e
-(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2
-(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda
-(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1
-c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2))
-(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2
-(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda
-(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1
-H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_:
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2
-(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1
-d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void)
-v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1
-x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T
-(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda
-(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2
-(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubv/defs.ma".
-
-include "LambdaDelta-1/T/props.ma".
-
-theorem csubv_bind_same:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall
-(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b)
-v2)))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b:
-B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1
-(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2:
-T).(csubv_bind c1 c2 H Abbr (\lambda (H0: (eq B Abbr Void)).(not_abbr_void
-H0)) Abbr v1 v2))) (\lambda (v1: T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst
-(sym_not_eq B Void Abst not_void_abst) Abst v1 v2))) (\lambda (v1:
-T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))).
-
-theorem csubv_refl:
- \forall (c: C).(csubv c c)
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n:
-nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda
-(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead
-c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t)))
-(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tlt/defs.ma".
-
-include "LambdaDelta-1/iso/defs.ma".
-
-include "LambdaDelta-1/clen/defs.ma".
-
-include "LambdaDelta-1/flt/defs.ma".
-
-include "LambdaDelta-1/app/defs.ma".
-
-include "LambdaDelta-1/cnt/defs.ma".
-
-include "LambdaDelta-1/cimp/defs.ma".
-
-include "LambdaDelta-1/csubv/defs.ma".
-
-include "LambdaDelta-1/subst/defs.ma".
-
-include "LambdaDelta-1/subst1/defs.ma".
-
-include "LambdaDelta-1/csubst1/defs.ma".
-
-include "LambdaDelta-1/fsubst0/defs.ma".
-
-include "LambdaDelta-1/next_plus/defs.ma".
-
-include "LambdaDelta-1/sty1/defs.ma".
-
-include "LambdaDelta-1/llt/defs.ma".
-
-include "LambdaDelta-1/aprem/defs.ma".
-
-include "LambdaDelta-1/ex0/defs.ma".
-
-include "LambdaDelta-1/wcpr0/defs.ma".
-
-include "LambdaDelta-1/csuba/defs.ma".
-
-include "LambdaDelta-1/nf2/defs.ma".
-
-include "LambdaDelta-1/ex2/defs.ma".
-
-include "LambdaDelta-1/csubc/defs.ma".
-
-include "LambdaDelta-1/pc1/defs.ma".
-
-include "LambdaDelta-1/ex1/defs.ma".
-
-include "LambdaDelta-1/csubt/defs.ma".
-
-include "LambdaDelta-1/wf3/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-include "LambdaDelta-1/lift/defs.ma".
-
-include "LambdaDelta-1/r/defs.ma".
-
-inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
-| drop_refl: \forall (c: C).(drop O O c c)
-| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e:
-C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u)
-e))))))
-| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c:
-C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d)
-(CHead c k (lift h (r k d) u)) (CHead e k u)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/drop/defs.ma".
-
-theorem drop_gen_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop
-h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x:
-C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c:
-C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d
-O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0:
-nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n))
-\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
-C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
-C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C
-c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq
-nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal
-nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda
-(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n))
-\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u:
-T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C
-(CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
-with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
-(CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O)
-(eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0:
-nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c
-e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq
-nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0
-(r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0)
-u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort
-_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
-(False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq
-nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))).
-
-theorem drop_gen_refl:
- \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
-\def
- \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O
-(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y:
-nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y
-n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0:
-nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0:
-nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to
-(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq
-nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c:
-C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq
-nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda
-(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S
-h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C
-(CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c
-e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c
-e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat
-h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in
-(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat
-n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h
-(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d)
-(\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let
-H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda
-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4)
-in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h
-H6)))))))))))))) y y0 x e H1))) H0))) H))).
-
-theorem drop_gen_drop:
- \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
-nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
-\def
- \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
-nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u)
-(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x))
-(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda
-(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to
-(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y
-x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_:
-nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x))))
-(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h))
-\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c
-c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq
-nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u)
-(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda
-(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c
-(CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda
-(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4:
-(((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to
-(drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S
-h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c
-k u))).(let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
-\Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let H9 \def
-(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
-(CHead c k u) H7) in ((let H10 \def (f_equal C T (\lambda (e0: C).(match e0
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H7) in (\lambda (H11: (eq K
-k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1:
-C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u))
-\to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda
-(c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0
-(\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c
-(CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def
-(eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17
-\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_:
-nat).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in
-(let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to
-((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h
-H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e))
-H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda
-(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3:
-(drop h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0
-d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c
-e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq
-nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead
-c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0
-(lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind
-nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq
-C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let
-H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h)
-H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
-\Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in
-((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
-(CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \def
-(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t:
-T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
-\Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false
-\Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0
-u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S
-h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h)
-(r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15:
-(eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S
-h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h)
-(r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1:
-C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0
-(\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19
-\def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1
-d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k
-H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c
-e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead
-e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h))
-\to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d)
-c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \def (eq_ind nat (S d)
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r
-k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11))))))))))))))))
-y1 y0 y x H2))) H1))) H0))) H)))))).
-
-theorem drop_gen_skip_r:
- \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
-(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda
-(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k
-d) e c)))))))))
-\def
- \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k
-u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0))
-(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d)
-u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0:
-(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y))
-(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x
-(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c)))))
-(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d))
-\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k
-(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda
-(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k
-u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C
-c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))))
-(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e
-k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3))))
-(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda
-(H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e
-(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0
-h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
-c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C
-e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d))
-\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
-(lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
-c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1:
-C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat
-O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2
-C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d)
-u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda
-(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
-C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
-(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0
-(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
-c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5:
-(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
-(CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let
-H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
-C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
-(CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda
-(H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t:
-T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead
-e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let
-H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1
-(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k
-d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12
-\def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in
-(let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
-((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
-(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9)
-in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12
-k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead
-c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0:
-C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0:
-nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow d0 |
-(S n) \Rightarrow n])) (S d0) (S d) H4) in (let H16 \def (eq_ind nat d0
-(\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to
-(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda
-(e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat
-d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d
-(\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n)
-u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
-c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u))
-(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c))
-c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9)))))
-u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))).
-
-theorem drop_gen_skip_l:
- \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
-(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T
-(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
-C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_:
-T).(drop h (r k d) c e))))))))))
-\def
- \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u)
-x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda
-(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e:
-C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0:
-(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x))
-(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e:
-C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v:
-T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
-d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq
-nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e:
-C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
-T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k
-d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda
-(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1:
-C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e:
-C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O
-(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind
-(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (CHead e k
-v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda
-(e: C).(\lambda (_: T).(drop O (r k d) c e)))) H4)) c0 H3)))) (\lambda (k0:
-K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r
-k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u))
-\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
-(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c
-e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq
-C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
-(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let
-H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
-C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
-(CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow
-u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H5) in
-(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
-C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to
-(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
-(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c
-H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e))
-H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d))
-\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v:
-T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r
-k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d)
-c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r
-k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda
-(e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda
-(v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_:
-T).(drop (S h0) (r k d) c e0)))) H15))))))))) H7)) H6))))))))))) (\lambda
-(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
-C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
-(S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda
-(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u
-(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
-e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda
-(H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0
-(lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K
-(\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0
-d0) u0)) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let
-rec lref_map (f: ((nat \to nat))) (d1: nat) (t: T) on t: T \def (match t with
-[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
-d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0)
-\Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in
-lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in
-(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
-C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u))
-\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0:
-C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def
-(eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13
-\def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9)
-in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
-((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C
-e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d)
-v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9)
-in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k
-H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda
-(v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
-T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
-(r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k
-d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0:
-C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
-T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
-(r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k
-d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C
-(CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
-(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
-e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat
-return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n]))
-(S d0) (S d) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat
-(r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T
-(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_:
-C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda
-(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19
-\def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in
-(eq_ind_r nat d (\lambda (n: nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v:
-T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq
-T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
-T).(drop h0 (r k d) c e0))))) (ex3_2_intro C T (\lambda (e0: C).(\lambda (v:
-T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq
-T (lift h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
-T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T
-(lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7))
-H6)))))))))))) h y0 y x H1))) H0))) H))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/drop/fwd.ma".
-
-include "LambdaDelta-1/lift/props.ma".
-
-include "LambdaDelta-1/r/props.ma".
-
-theorem drop_skip_bind:
- \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
-d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b)
-(lift h d u)) (CHead e (Bind b) u))))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
-(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b)
-d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e
-(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
-
-theorem drop_skip_flat:
- \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
-(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat
-f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
-(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat
-f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead
-e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S
-d))))))))).
-
-theorem drop_S:
- \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
-nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
-C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to
-(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
-T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b)
-u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O
-O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort
-n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O
-(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead
-e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
-with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
-(CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1))))
-(drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda
-(H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e
-(Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n:
-nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead
-c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b)
-u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
-\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead
-c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u)
-(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def
-(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t)
-(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0))
-in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0
-(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b)
-(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0
-c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_:
-(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0
-k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b)
-u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0:
-nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b)
-u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
-
-theorem drop_ctail:
- \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop
-h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1)
-(CTail k u c2))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u:
-T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda
-(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n)
-c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h
-O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0:
-(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d
-O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n))
-(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u
-(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O
-(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2
-H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2:
-C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
-nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k
-u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3:
-C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n
-(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u
-(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n:
-nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop
-n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O
-(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t)
-(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
-(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3
-H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to
-(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail
-k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0:
-K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O
-(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n:
-nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to
-(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail
-k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t)
-c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda
-(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
-(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e)))
-(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2:
-(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4
-\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k
-t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2
-k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k
-x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u
-c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n
-(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop
-h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4
-(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0:
-T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1))))
-(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u)
-x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d)))))))
-c1).
-
-theorem drop_mono:
- \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h
-d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0
-x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d:
-nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2:
-C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq
-nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort
-n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C
-x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1
-(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r
-C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h
-(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d
-(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda
-(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1))))
-(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda
-(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h:
-nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1
-x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d:
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t)
-x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2))))))
-(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1)
-\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2)))))
-(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1:
-(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C
-x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t)))
-(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2
-(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_:
-(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t)
-x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t)
-x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O
-(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n
-H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
-(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq
-C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t)
-x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t)
-x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
-C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0:
-C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4:
-(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0
-x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
-C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4:
-C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7:
-(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r
-C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1
-(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to
-(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0
-(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1
-(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0:
-nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop
-h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k
-n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r
-k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda
-(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0))
-(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n)
-t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n)
-H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k
-x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h
-H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n)
-H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1)))))))
-(drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c).
-
-theorem drop_conf_lt:
- \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall
-(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h:
-nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop
-h (r k d) c0 e0)))))))))))))
-\def
- \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u:
-T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
-(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u:
-T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k
-u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop
-h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S
-(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in
-(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v))))
-(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
-T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda
-(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d))
-x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k
-x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
-h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T
-(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda
-(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda
-(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v:
-T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k
-d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0
-e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall
-(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d))
-c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
-v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda
-(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0)
-O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
-nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S
-i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind
-(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
-T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort
-n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d))
-O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee
-in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda
-(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop
-(S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
-c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1))))))))
-(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0)
-d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k
-d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda
-(k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1
-t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
-nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
-(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e:
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
-d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
-T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
-(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
-T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
-(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b)
-(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1
-x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b)
-c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T
-u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O
-(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7:
-(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k
-x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0
-x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h
-(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda
-(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e:
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
-d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
-T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
-(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
-T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
-(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f)
-(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1
-x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S
-i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
-c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
-v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1)
-(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))
-(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d)
-x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h
-(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead
-x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r
-k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1)
-H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e
-H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2)))))))))
-k0)))) c)))))) i)).
-
-theorem drop_conf_ge:
- \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
-(plus d h) i) \to (drop (minus i h) O e a)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c:
-C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e
-a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
-d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda
-(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def
-(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus
-O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5
-\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def
-(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O
-(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0:
-C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h
-H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0:
-nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
-(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a:
-C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d
-h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n:
-nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2:
-(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d
-O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda
-(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n))
-(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6:
-(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O
-O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2
-O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S
-i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0)
-O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0
-a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n)
-c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n))
-H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0)))))
-(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S
-i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
-d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e
-a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S
-i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop
-(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1:
-(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t)
-e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h
-n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S
-i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda
-(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0
-(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e
-a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le
-(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop
-(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b)
-c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda
-(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O
-h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0)
-O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a
-c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e
-t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0:
-nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h)
-(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
-(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
-i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind
-b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0)
-v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop
-(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
-e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0)
-x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0
-(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S
-(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a))
-(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t
-i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h)
-(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e
-H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3)))))))))
-(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat
-f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
-(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S
-i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le
-(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h
-O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind
-(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S
-i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0
-(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0
-(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat
-f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead
-c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead
-c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O
-e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7:
-(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S
-h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0:
-nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h)
-(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
-(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
-i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat
-f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0)
-v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop
-(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
-e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0)
-x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0
-(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def
-(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0
-(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h))
-(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in
-(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f)
-x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h)
-(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e
-H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3)))))))))
-k)))) c))))) i).
-
-theorem drop_conf_rev:
- \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
-(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1:
-C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
-\def
- \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2:
-C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2)
-\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1
-e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1
-e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let
-H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl
-e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1:
-C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda
-(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2:
-C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O
-c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind
-(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2:
-C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n:
-nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda
-(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2
-(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1:
-(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O
-O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n)
-H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
-True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2))
-(\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S
-j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop
-(S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to
-(ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S
-j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda
-(H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i:
-nat).(\lambda (H0: (drop i O c2 e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0)
-O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
-C).(drop i (S j0) c1 (CHead e2 k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop
-(r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2
-\def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1:
-C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda
-(c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda
-(H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda
-(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2
-(Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind
-b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4
-t))))) H2))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2
-e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C
-(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1
-e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i
-(S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0)
-O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1:
-C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat
-f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0
-x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t)))))
-H2))))) k (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j).
-
-theorem drop_trans_le:
- \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h
-(minus d i) e1 e2)))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to
-(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to
-(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1
-e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d:
-nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
-nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O
-c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2
-(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda
-(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C
-(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1
-(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0:
-nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1:
-C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
-C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda
-(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind
-(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2:
-C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
-C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0)
-O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h
-O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat
-(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C
-(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S
-i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_:
-(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow
-False])) I (S x) H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1
-e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H4))))) (le_gen_S i0
-O H))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall
-(c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall
-(e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1
-e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda
-(H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
-C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0)
-O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1:
-C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda
-(c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n)
-c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2
-(CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S
-i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))
-(\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
-(eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c
-e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq
-nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1:
-C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort
-n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C
-(CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n)
-e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def
-(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in
-(False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
-(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6))))
-(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0))))))))
-(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h
-(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda
-(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0))
-e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
-T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to
-(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S
-i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
-e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h:
-nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
-C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e:
-C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda
-(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_:
-T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O
-(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
-e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0
-(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda
-(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c:
-C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h
-(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O
-(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
-e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop
-h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2
-(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S
-d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda
-(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0)
-O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop
-h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r
-(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2
-(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3
-t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3:
-C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t)
-c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T
-(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_:
-C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e:
-C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1:
-C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus
-(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C
-c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0)
-x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3
-(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in
-(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1:
-C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus
-(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1))
-(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1:
-C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1))
-(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x:
-C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S
-i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
-(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S
-i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1))
-H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3)))))))
-(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
-
-theorem drop_trans_ge:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2
-e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2:
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
-C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2))))))))))
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
-nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
-c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h)
-O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d
-(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2
-H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall
-(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall
-(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1
-e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2:
-C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c
-e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
-nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0:
-(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort
-n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda
-(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d
-O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n)
-e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4)
-in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort
-n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop
-(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda
-(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n)
-(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind
-nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop)
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind
-(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n
-(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2:
-C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
-nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d
-(S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda
-(t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall
-(h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O
-c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t)
-e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k
-t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to
-(drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead
-c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda
-(_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O
-c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat
-i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2
-(drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda
-(n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2:
-C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O
-(CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t)
-c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le
-O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O
-(CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k
-(plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r
-k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n)
-(drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0
-n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm
-i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop
-h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le
-d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h:
-nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2:
-C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S
-i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k
-v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda
-(e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O
-(CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3
-(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4:
-(drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall
-(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O
-c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t)
-e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c:
-C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t
-(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k
-x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S
-i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k
-d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S
-(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind
-(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to
-(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r
-(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0
-(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f:
-F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat
-f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4
-(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3)))))))))
-(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/drop/defs.ma".
-
-include "LambdaDelta-1/lift1/defs.ma".
-
-inductive drop1: PList \to (C \to (C \to Prop)) \def
-| drop1_nil: \forall (c: C).(drop1 PNil c c)
-| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds
-c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))).
-
-definition ptrans:
- PList \to (nat \to PList)
-\def
- let rec ptrans (hds: PList) on hds: (nat \to PList) \def (\lambda (i:
-nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow
-(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d)
-with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow
-q])))])) in ptrans.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/drop1/defs.ma".
-
-theorem drop1_gen_pnil:
- \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq
-PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1
-c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda
-(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c
-c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c)))
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds:
-PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to
-(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def
-(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee in PList return
-(\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _)
-\Rightarrow True])) I PNil H4) in (False_ind (eq C c3 c5) H5)))))))))))) y c1
-c2 H0))) H))).
-
-theorem drop1_gen_pcons:
- \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h:
-nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda
-(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))
-\def
- \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq
-PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_:
-PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds
-c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind
-(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d
-hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1
-hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d
-hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee in
-PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _
-_ _) \Rightarrow False])) I (PCons h d hds) H1) in (False_ind (ex2 C (\lambda
-(c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 hds c2 c))) H2)))) (\lambda
-(c2: C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1:
-(drop h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2:
-(drop1 hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2
-C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6
-c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let H5
-\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
-(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
-(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat
-(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
-[PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0)
-(PCons h d hds) H4) in ((let H7 \def (f_equal PList PList (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
-\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h
-d hds) H4) in (\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let
-H10 \def (eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds))
-\to (ex2 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6
-c5))))) H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p:
-PList).(drop1 p c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda
-(n: nat).(drop h0 n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda
-(n: nat).(drop n d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h
-d c2 c6)) (\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6))
-H5)))))))))))) y c1 c3 H0))) H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/drop1/fwd.ma".
-
-include "LambdaDelta-1/getl/drop.ma".
-
-theorem drop1_getl_trans:
- \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
-\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
-i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i)
-e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1
-(ptrans hds i) v)))))))))))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
-C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
-C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
-(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl
-(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v))))))))))))))
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda
-(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in
-(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1))
-(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda
-(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b)
-v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d:
-nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2:
-C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v:
-T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda
-(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i)
-c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda
-(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl
-i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d
-H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3))
-(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match
-(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1))
-(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow
-(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
-(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
-hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4:
-(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0:
-bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons
-h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
-hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow
-(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
-(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v))))))
-(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0
-i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow
-(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow
-(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true
-\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
-(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d
-(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)])
-v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def
-(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2:
-C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x
-(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2:
-C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1))
-(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0:
-C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans
-hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def
-(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b
-x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda
-(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans
-hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S
-(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S
-(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0
-i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans
-hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2
-(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0
-i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1
-x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0
-i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead
-e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i))
-v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0
-i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d)
-false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in
-(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2:
-C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))
-(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl
-(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))))
-(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8:
-(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let
-H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b)
-(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1
-(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2
-(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans
-hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/drop1/fwd.ma".
-
-include "LambdaDelta-1/drop/props.ma".
-
-include "LambdaDelta-1/getl/defs.ma".
-
-theorem drop1_skip_bind:
- \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
-C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b)
-(lift1 hds u)) (CHead e (Bind b) u)))))))
-\def
- \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p:
-PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p)
-(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def
-(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead
-c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c
-H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
-(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead
-c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda
-(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def
-(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda
-(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S
-n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))
-(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x
-e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b)
-(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e
-(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))).
-
-theorem drop1_cons_tail:
- \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop
-h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to
-(drop1 (PConsTail hds h d) c1 c3))))))))
-\def
- \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda
-(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
-c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def
-(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d
-PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y))))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0:
-((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
-c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x
-\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C
-(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1
-(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n
-n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3
-(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))).
-
-theorem drop1_trans:
- \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0)
-\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1
-(papp is1 is2) c1 c2)))))))
-\def
- \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
-C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2:
-C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1:
-C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2:
-PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def
-(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c:
-C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0:
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0:
-C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0
-c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0:
-C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda
-(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0
-p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1
-c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1
-c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x
-c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1)))))
-H2))))))))))))) is1).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/A/defs.ma".
-
-include "LambdaDelta-1/G/defs.ma".
-
-definition gz:
- G
-\def
- mk_G S lt_n_Sn.
-
-inductive leqz: A \to (A \to Prop) \def
-| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
-(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort
-h2 n2))))))
-| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3:
-A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ex0/defs.ma".
-
-include "LambdaDelta-1/leq/defs.ma".
-
-include "LambdaDelta-1/aplus/props.ma".
-
-theorem aplus_gz_le:
- \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A
-(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n))))))
-\def
- \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0:
-nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h)
-n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y
-\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n)
-(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0:
-nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A
-(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h:
-nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A
-(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O
-\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n:
-nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n))
-k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O
-(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0
-n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n)))
-(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0:
-nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S
-(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0))
-(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz
-(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n:
-nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz
-(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S
-k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda
-(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz
-(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0)
-k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a:
-A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0)
-k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S
-n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k).
-
-theorem aplus_gz_ge:
- \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A
-(aplus gz (ASort h n) k) (ASort (minus h k) n)))))
-\def
- \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h:
-nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0)
-n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0:
-nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O)
-(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0
-h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda
-(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz
-(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le
-(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0:
-nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n))
-(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0
-x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O
-n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0)
-n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0))
-n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in
-(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus
-gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n))
-k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus
-gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc
-gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)).
-
-theorem next_plus_gz:
- \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
-\def
- \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat
-(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0:
-nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat
-S (next_plus gz n n0) (plus n0 n) H))) h)).
-
-theorem leqz_leq:
- \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz
-(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda
-(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda
-(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k
-h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2
-(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def
-(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort
-h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1
-(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k)
-(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2)
-(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat)
-on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O |
-(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow
-(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2
-k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in
-(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n:
-nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n:
-nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal
-nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1))
-(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2
-k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a
-(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1
-(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort
-h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus
-(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat
-(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2)
-n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind
-A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return
-(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])
-| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in
-(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1
-k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k
-h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A
-a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1))
-(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2)
-k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort
-(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in
-(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k)
-n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A
-(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k)))
-(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k)))
-n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort
-n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow
-False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1
-n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A
-(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2)
-k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4
-\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O
-(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le
-k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
-plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus
-(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in
-(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3:
-A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4:
-A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4
-a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
-
-theorem leq_leqz:
- \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind
-(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda
-(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus
-h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A
-(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1)))
-(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A
-(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2)))
-(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus
-(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus
-h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2))
-(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n:
-nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus
-h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq
-A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz
-n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1
-h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n:
-nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus
-h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal
-nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2
-n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2)
-(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2)
-(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2)
-(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus
-(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2))
-(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1
-h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0:
-A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0
-a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda
-(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-definition ex1_c:
- C
-\def
- CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O).
-
-definition ex1_t:
- T
-\def
- THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ex1/defs.ma".
-
-include "LambdaDelta-1/ty3/fwd.ma".
-
-include "LambdaDelta-1/pc3/fwd.ma".
-
-include "LambdaDelta-1/nf2/pr3.ma".
-
-include "LambdaDelta-1/nf2/props.ma".
-
-include "LambdaDelta-1/arity/defs.ma".
-
-include "LambdaDelta-1/leq/props.ma".
-
-theorem ex1__leq_sort_SS:
- \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc
-g (asucc g (ASort (S (S k)) n))))))
-\def
- \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g
-(asucc g (ASort (S (S k)) n)))))).
-
-theorem ex1_arity:
- \forall (g: G).(arity g ex1_c ex1_t (ASort O O))
-\def
- \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S
-(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O))
-(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)
-O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g
-(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O))
-(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O))
-O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead
-(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst)
-(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort
-O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort
-O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort
-O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O))
-O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))).
-
-theorem ex1_ty3:
- \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P:
-Prop).P)))
-\def
- \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort
-O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t:
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind
-Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst)
-u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1))
-u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef
-(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda
-(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O
-t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O
-t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O
-x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind
-Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2:
-T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S
-O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_:
-T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort
-O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0
-x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
-x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef
-(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_:
-T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O)))
-O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
-t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9:
-T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9)
-x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7
-(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
-x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7
-(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S
-O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x:
-C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr)
-x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
-[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) |
-(Flat _) \Rightarrow False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
-x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2
-(Bind Abbr) x3) H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda
-(H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0)
-x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
-T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
-x7 (Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def
-(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
-Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
-C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P
-(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7
-(Bind Abst) x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H17)))))
-H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O))
-H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
-(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (\lambda (H3: (ex3_3 C T
-T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2:
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: (pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex3_2_ind T T
-(\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind
-Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_:
-T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t)))
-(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind
-Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: T).(\lambda (x6:
-T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S
-O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g (CHead (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C
-T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst)
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P
-(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
-T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
-T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
-x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def
-(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
-Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
-C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P
-(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7
-(Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _)
-\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C
-T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3)
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O)
-(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda
-(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
-(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def
-(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def
-(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead
-(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst)
-(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort
-O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CSort O)
-(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr)
-x8) (TSort O) H23)) in (False_ind P H24)))))))) H17))))) H14)))))))) H10))
-(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0)
-x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O))
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8:
-T).(\lambda (x9: T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S
-O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
-x7 (Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def
-(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind
-Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e:
-C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P
-(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7
-(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _)
-\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O))
-(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C
-T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3)
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O)
-(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda
-(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6
-(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def
-(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def
-(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead
-(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst)
-(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort
-O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match
-e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x7 | (CHead c _
-_) \Rightarrow c])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst)
-(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O)
-H23)) in ((let H25 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow x8 | (CHead _ _ t) \Rightarrow
-t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O))
-(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in
-(\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind T x8 (\lambda (t:
-T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def (eq_ind T x8 (\lambda
-(t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)) H11 (TSort O)
-H25) in (let H29 \def (eq_ind C x7 (\lambda (c: C).(ty3 g c (TSort O) x9))
-H27 (CSort O) H26) in (or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_:
-T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_:
-C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H30:
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O
-t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind
-Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0
-t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift
-(S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
-e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g
-e u0 t)))) P (\lambda (x10: C).(\lambda (x11: T).(\lambda (x12: T).(\lambda
-(_: (pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (lift (S O) O x12) x4)).(\lambda (H32: (getl O (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr)
-x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def (eq_ind C (CHead x10
-(Bind Abbr) x11) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst
-(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O)
-(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30))
-(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
-T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11:
-T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32:
-(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow x10 | (CHead c _ _) \Rightarrow c])) (CHead x10
-(Bind Abst) x11) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O))
-(CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11)
-H32))) in ((let H35 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow x11 | (CHead _ _ t)
-\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O)
-(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead
-x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O)
-(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g
-x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t:
-T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10
-(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst)
-(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
-x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind
-b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S
-O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O)
-O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
-t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O
-(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O)))
-(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort
-O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O
-(S O) O (le_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
-(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind
-Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda
-(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef
-O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort
-O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)
-(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst)
-(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O)
-(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead
-(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort
-O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O))
-x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst)
-(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind
-Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O)
-(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T
-(TLRef (S (S O))) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef n) \Rightarrow (match n
-in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S n0)
-\Rightarrow (match n0 in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])]) | (THead _ _ _) \Rightarrow
-False])) I (TLRef (S O)) H46) in (False_ind P H47)))))) H43)))))
-(pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 x5 x1 H7)))))))
-H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) (Bind Abst) (TSort
-O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) H17))))) H14))))))))
-H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) H8)))))))
-(ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O))
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) (TSort O)
-(THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead (CHead
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst)
-(TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort O)
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef
-O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-definition ex2_c:
- C
-\def
- CSort O.
-
-definition ex2_t:
- T
-\def
- THead (Flat Appl) (TSort O) (TSort O).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ex2/defs.ma".
-
-include "LambdaDelta-1/nf2/defs.ma".
-
-include "LambdaDelta-1/pr2/fwd.ma".
-
-include "LambdaDelta-1/arity/fwd.ma".
-
-theorem ex2_nf2:
- nf2 ex2_c ex2_t
-\def
- \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O)
-(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2
-H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
-O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
-(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
-Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
-O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat
-Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O)
-(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2
-(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O)
-x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1
-(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O)
-(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t:
-T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort
-(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O))
-(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal
-T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1:
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
-(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T
-(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead
-(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2
-x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b:
-B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def
-(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O)
-(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O)
-x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7
-\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in
-(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr)
-(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort
-O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O)
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O)
-(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2)
-(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
-(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq
-T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0)
-x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O)
-(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead
-(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in
-(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead
-(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O
-H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
-(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O))
-t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1
-x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2
-H8))))))))))))))) H1)) H0))).
-
-theorem ex2_arity:
- \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P:
-Prop).P)))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat
-Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def
-(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda
-(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O)
-(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O)
-(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let
-H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O
-(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a
-a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P
-(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_:
-(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def
-(eq_ind A (ASort O O) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H6) in (False_ind P H7))))))) H3)))))) H0))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/C/defs.ma".
-
-definition fweight:
- C \to (T \to nat)
-\def
- \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))).
-
-definition flt:
- C \to (T \to (C \to (T \to Prop)))
-\def
- \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt
-(fweight c1 t1) (fweight c2 t2))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/flt/defs.ma".
-
-include "LambdaDelta-1/C/props.ma".
-
-theorem flt_thead_sx:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
-(THead k u t)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
-T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u)
-(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u)
-(tweight t)) (le_plus_l (tweight u) (tweight t))))))).
-
-theorem flt_thead_dx:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c
-(THead k u t)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
-T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u)
-(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u)
-(tweight t)) (le_plus_r (tweight u) (tweight t))))))).
-
-theorem flt_shift:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c
-k u) t c (THead k u t)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat
-(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt
-(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus
-(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
-(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
-c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
-t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
-(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
-(tweight t))))))).
-
-theorem flt_arith0:
- \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t
-(CHead c k t) (TLRef i)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_:
-nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
-
-theorem flt_arith1:
- \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle
-(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i:
-nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
-\def
- \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda
-(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_:
-K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1)
-(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H
-(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
-nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
-(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
-(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S
-O))))))))))).
-
-theorem flt_arith2:
- \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1
-t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt
-c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda
-(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
-(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
-(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
-t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S
-O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
-
-theorem flt_trans:
- \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1
-t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt
-c1 t1 c3 t3))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3:
-T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1
-t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))).
-
-theorem flt_wf__q_ind:
- \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C
-\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq
-nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall
-(t: T).(P c t))))
-\def
- let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
-(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
-(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c:
-C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c:
-C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
-
-theorem flt_wf_ind:
- \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2:
-T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
-\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
-\def
- let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
-(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
-(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2:
-T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
-\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda
-(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m:
-nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda
-(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1:
-nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq
-nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0
-(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2
-(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c
-t))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubst0/defs.ma".
-
-inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def
-| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1
-t2))
-| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2
-t1))
-| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2:
-C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/fsubst0/defs.ma".
-
-theorem fsubst0_gen_base:
- \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall
-(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1
-c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0
-i v t1 t2) (csubst0 i v c1 c2)))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind
-i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v
-t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0
-i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0
-(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1
-c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0
-i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i
-v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1
-t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj
-(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0:
-T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0
-i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1
-t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj
-(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/props.ma".
-
-include "LambdaDelta-1/clear/drop.ma".
-
-theorem clear_getl_trans:
- \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to
-(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3:
-C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1
-c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2
-c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1
-(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda
-(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to
-(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2:
-C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall
-(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda
-(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda
-(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1
-c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3)
-\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t)
-c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda
-(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to
-(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c
-(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def
-(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in
-(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3))
-(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda
-(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n
-H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f)
-t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1
-c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i).
-
-theorem getl_clear_trans:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to
-(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3))))))
-\def
- \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1
-c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all
-c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e:
-C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1
-x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in
-(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2
-(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6
-\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5)
-in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind
-x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1
-c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0
-x1 c3 x2 H7)))))))) H4))))) H1))))))).
-
-theorem getl_clear_bind:
- \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c
-(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2)
-\to (getl (S n) c e2))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1:
-C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2:
-C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda
-(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n)
-(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_:
-(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0)
-(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1:
-C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2:
-C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear
-(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n:
-nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0
-k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda
-(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b)
-v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1]))
-(CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1
-(Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e in
-C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _)
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0
-(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead e1
-(Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b)
-v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8
-\def (eq_ind C e1 (\lambda (c1: C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b
-(\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b)
-n c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: F).(\lambda (H2: (clear
-(CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v
-(clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))) k
-H0))))))))))) c)).
-
-theorem getl_clear_conf:
- \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to
-(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3:
-C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2
-c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1
-c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c:
-C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in
-(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3
-(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3
-\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead
-x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c:
-C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3
-(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall
-(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2)
-\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall
-(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n)
-c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n)
-(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0)
-c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c:
-C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2:
-C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda
-(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0:
-K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl
-(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b)
-t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c
-(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3
-(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4)))))
-(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda
-(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n
-H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/props.ma".
-
-theorem getl_dec:
- \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda
-(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d:
-C).((getl i c d) \to (\forall (P: Prop).P)))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T
-(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b)
-v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P))))))
-(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e:
-C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b)
-v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P)))
-(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P:
-Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall
-(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
-T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to
-(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i:
-nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall
-(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind
-(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
-T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O
-(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl
-(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead
-c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0
-(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e:
-C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e
-(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def
-(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d:
-C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e:
-C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e
-(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to
-(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda
-(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T
-(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b)
-v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl
-O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O
-(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0:
-C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind
-x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
-T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d:
-C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro
-C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat
-f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2)
-O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to
-(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v))))))
-(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))
-(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P:
-Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t
-(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n:
-nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
-(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d:
-C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H
-(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda
-(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall
-(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T
-(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t)
-(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to
-(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda
-(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind
-C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead
-e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
-(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d:
-C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0:
-C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0
-(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v))))))
-(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))
-(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n)
-(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0
-(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n)
-c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e:
-C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind
-b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P:
-Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t)
-d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1)))))
-i)))))) c).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/drop/defs.ma".
-
-include "LambdaDelta-1/clear/defs.ma".
-
-inductive getl (h: nat) (c1: C) (c2: C): Prop \def
-| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h
-c1 c2))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/props.ma".
-
-include "LambdaDelta-1/clear/drop.ma".
-
-theorem getl_drop:
- \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
-nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
-C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to
-(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
-T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b)
-u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n)
-e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u:
-T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0
-e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
-T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t)
-(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0:
-(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear
-(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e)))
-(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind
-b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow
-c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0
-(CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b |
-(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
-[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u)
-(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in
-((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e
-(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e
-c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t)
-c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0))
-(drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2))))
-(\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b)
-u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind
-b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl
-e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n:
-nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S
-n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e
-(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0:
-nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t
-n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
-
-theorem getl_drop_conf_lt:
- \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i:
-nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h:
-nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
-h d c0 e0)))))))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1:
-C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d))
-c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
-(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i
-(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i
-(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda
-(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i:
-nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h:
-nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
-h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t)
-(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def
-(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C
-(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0
-(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x:
-C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead
-c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to
-((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead
-e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))
-(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda
-(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind
-b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
-(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda
-(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u))
-\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
-(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda
-(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda
-(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1:
-K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0)
-(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0:
-B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda
-(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind
-b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0
-H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow
-(match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
-(Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0)
-(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def
-(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind
-b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0
-H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14
-\def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind
-b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i
-O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def
-(eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2
-(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in
-(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead
-c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0:
-C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead
-e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))
-(\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b)
-d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20:
-(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1:
-T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda
-(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1)
-H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
-T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda
-(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1))
-(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind
-b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17
-e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O
-(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat
-f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n
-d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0))
-\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
-(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
-h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t)
-e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda
-(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u))
-\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda
-(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C
-(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0
-(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in
-((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1]))
-(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0
-(Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1)
-\Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead
-c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat
-f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2:
-C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b)
-u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O
-(CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def
-(eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0
-(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f)
-H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d))
-(CHead c0 k1 t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0:
-C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda
-(v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0:
-C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2:
-T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t
-(lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f)
-(plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
-(r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2:
-C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u))
-\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda
-(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2)
-H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda
-(t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind
-b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0
-(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24
-(lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1
-(Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
-h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
-v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead
-e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))
-(\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d
-x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop
-h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0
-(Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda
-(v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d
-x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1
-(Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda
-(t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0
-(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))
-(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h
-d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2)
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))
-x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28
-f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus
-O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi:
-(((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t)
-(CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0
-(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind
-b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T
-C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus
-(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t)
-(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0)
-\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda
-(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0)
-e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
-e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0
-k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d))
-v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0)))
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
-(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2:
-T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r
-k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0
-x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S
-i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S
-(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0
-(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1
-(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
-v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let
-H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to
-((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
-e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda
-(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
-(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T
-t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead
-c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
-h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2)
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1
-e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind
-T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k
-x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0
-O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S
-(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d))
-(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S
-i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h
-(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k
-i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0)
-(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1
-(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d
-H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
-v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4:
-C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1
-(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def
-(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S
-(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3)
-H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind
-b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22)
-in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
-T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v:
-T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift
-h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22))))))))
-H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k
-H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)).
-
-theorem getl_drop_conf_ge:
- \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d
-h) i) \to (getl (minus i h) e a)))))))))
-\def
- \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
-d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H)
-in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0
-a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c
-x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i
-x c H3 e h d H0 H1) H4)))) H2)))))))))).
-
-theorem getl_conf_ge_drop:
- \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i:
-nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1
-c2) \to (drop i O c2 e))))))))
-\def
- \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda
-(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O))
-(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e
-u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S
-i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in
-H3)))))))).
-
-theorem getl_drop_conf_rev:
- \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
-(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i
-c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2))
-(\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
-\def
- \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1
-e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2
-H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
-
-theorem drop_getl_trans_lt:
- \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2:
-C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda
-(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda
-(e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
-\def
- \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
-c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i
-c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b)
-v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e:
-C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead
-e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d
-(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4:
-(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1
-e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1:
-C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1:
-C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O
-c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat
-(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i)))
-(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b
-e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h
-(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C
-(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v))))
-(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda
-(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda
-(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i
-c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h
-(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus
-d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S
-(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))).
-
-theorem drop_getl_trans_le:
- \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2
-e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0)))
-(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_:
-C).(\lambda (e1: C).(clear e1 e2))))))))))))
-\def
- \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1:
-C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
-c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def
-(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
-(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_:
-C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i)
-e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x:
-C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def
-(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i
-O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda
-(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1:
-C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1
-e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h
-(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i
-O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
-(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5)))))
-H2)))))))))).
-
-theorem drop_getl_trans_ge:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2)
-\to ((le d i) \to (getl (plus i h) c1 e2)))))))))
-\def
- \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d:
-nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2:
-C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def
-(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
-(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x:
-C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro
-(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
-
-theorem getl_drop_trans:
- \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to
-(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i
-h)) O c1 e2)))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h:
-nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2
-e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2:
-C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2:
-C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2
-H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda
-(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2:
-C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2
-e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall
-(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2:
-C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead
-c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3:
-C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b)
-t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop
-(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead
-c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S
-i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2))
-H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2
-(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead
-c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2
-t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_:
-(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i:
-nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t)
-e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
-C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S
-i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop
-(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2
-i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f:
-F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n:
-nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i:
-nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t)
-e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2:
-C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f)
-(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2)
-(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0)
-t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to
-(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i
-n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2
-(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i)
-O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S
-(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/fwd.ma".
-
-include "LambdaDelta-1/clear/props.ma".
-
-include "LambdaDelta-1/flt/props.ma".
-
-theorem getl_flt:
- \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i:
-nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
-C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to
-(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b)
-u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef
-i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u:
-T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef
-i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
-T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t)
-(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0:
-(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear
-(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef
-O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e
-(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _)
-\Rightarrow c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind
-b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b |
-(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
-[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u)
-(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in
-((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e
-(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e
-c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef
-O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef
-O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef
-O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2)))) (\lambda
-(f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b)
-u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u)
-(clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) k
-(getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n:
-nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u
-(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead
-e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind
-b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/defs.ma".
-
-include "LambdaDelta-1/drop/fwd.ma".
-
-include "LambdaDelta-1/clear/fwd.ma".
-
-theorem getl_gen_all:
- \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2
-C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1
-c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e:
-C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda
-(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda
-(e0: C).(clear e0 c2)) e H0 H1)))) H)))).
-
-theorem getl_gen_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to
-(\forall (P: Prop).P))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h
-(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H)
-in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear
-e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2:
-(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P
-(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
-(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2
-(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0
-H1))))) H0)))))).
-
-theorem getl_gen_O:
- \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x)))
-\def
- \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def
-(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0))
-(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1:
-(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0
-(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))).
-
-theorem getl_gen_S:
- \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
-nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x))))))
-\def
- \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
-nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all
-(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c
-k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0:
-C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0
-x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))).
-
-theorem getl_gen_2:
- \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3
-B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind
-b) v)))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1
-c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e:
-C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b:
-B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v))))))
-(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x
-c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B
-C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind
-b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4:
-(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c:
-C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind
-x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda
-(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b:
-B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c
-(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4))))))
-H3))))) H0))))).
-
-theorem getl_gen_flat:
- \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d))))))
-\def
- \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n
-e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e
-(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d
-H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to
-(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v)
-d)).(getl_gen_S (Flat f) e d v n H0)))) i))))).
-
-theorem getl_gen_bind:
- \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d
-(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda
-(j: nat).(getl j e d)))))))))
-\def
- \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or
-(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j:
-nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl
-O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c:
-C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda
-(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl
-(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat
-(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e
-(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind
-b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d
-(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n:
-nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n
-O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S
-j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e
-(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b)
-v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl
-j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j:
-nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n
-H0)))))) i))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/drop.ma".
-
-include "LambdaDelta-1/getl/clear.ma".
-
-theorem getl_conf_le:
- \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
-(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e
-a))))))))
-\def
- \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda
-(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda
-(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e
-a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x
-e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1)
-e H4)))) H2))))))))).
-
-theorem getl_trans:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl
-h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1
-e2)))))))
-\def
- \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
-nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2
-e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e:
-C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2)
-(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x
-e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1
-e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c:
-C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O
-h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to
-(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def
-(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x
-H_y H3))))) i H2)))) H1)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/getl/fwd.ma".
-
-include "LambdaDelta-1/drop/props.ma".
-
-include "LambdaDelta-1/clear/props.ma".
-
-theorem getl_refl:
- \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u)
-(CHead c (Bind b) u))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind
-b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b)
-u)) (clear_bind b c u)))).
-
-theorem getl_head:
- \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k
-h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e))))))
-\def
- \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
-(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k
-h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0:
-C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1:
-(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k
-u) e x (drop_drop k h c x H1 u) H2)))) H0))))))).
-
-theorem getl_flat:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall
-(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c
-e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in
-(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e))
-(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c
-x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to
-(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def
-(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in
-(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c
-(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_:
-(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop
-(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat
-f) h0 c x H3 u) H2)))) h H1)))) H0))))))).
-
-theorem getl_ctail:
- \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
-nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v:
-T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda
-(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C
-(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b)
-u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x:
-C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b)
-u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v
-x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))).
-
-theorem getl_mono:
- \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to
-(\forall (x2: C).((getl h c x2) \to (eq C x1 x2))))))
-\def
- \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c
-x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all
-c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e:
-C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c
-x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in
-(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq
-C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear
-x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0
-(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0:
-C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def
-(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0
-H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-inductive iso: T \to (T \to Prop) \def
-| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2)))
-| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2)))
-| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2:
-T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/iso/defs.ma".
-
-include "LambdaDelta-1/tlist/defs.ma".
-
-theorem iso_gen_sort:
- \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda
-(n2: nat).(eq T u2 (TSort n2))))))
-\def
- \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1)
-u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex
-nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0:
-(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1))
-\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0:
-nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2
-\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
-with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow n0])) (TSort n0) (TSort n1) H1) in (ex_intro nat (\lambda (n3:
-nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort n2))))))) (\lambda
-(i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TSort n1))).(let
-H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex nat
-(\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1:
-T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
-K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T
-(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TSort n1) H1) in (False_ind (ex nat (\lambda (n2:
-nat).(eq T (THead k v2 t2) (TSort n2)))) H2)))))))) y u2 H0))) H))).
-
-theorem iso_gen_lref:
- \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda
-(n2: nat).(eq T u2 (TLRef n2))))))
-\def
- \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1)
-u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex
-nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0:
-(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1))
-\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0:
-nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2
-\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (TLRef n1) H1) in (False_ind (ex nat
-(\lambda (n3: nat).(eq T (TSort n2) (TLRef n3)))) H2))))) (\lambda (i1:
-nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2
-\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
-with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _)
-\Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in (ex_intro nat (\lambda (n2:
-nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))))))) (\lambda
-(v1: T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
-K).(\lambda (H1: (eq T (THead k v1 t1) (TLRef n1))).(let H2 \def (eq_ind T
-(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2:
-nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))).
-
-theorem iso_gen_head:
- \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso
-(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
-(THead k v2 t2)))))))))
-\def
- \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda
-(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t:
-T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2:
-T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y
-u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to
-(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2))))))))
-(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k
-v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H1)
-in (False_ind (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T (TSort n2)
-(THead k v2 t2))))) H2))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda
-(H1: (eq T (TLRef i1) (THead k v1 t1))).(let H2 \def (eq_ind T (TLRef i1)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda (v2:
-T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) (\lambda
-(v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (k0:
-K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
-\Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H3 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
-\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in (\lambda (_: (eq T
-v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(ex_2 T T
-(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k1 v2 t2) (THead k v3 t3))))))
-(ex_2_intro T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2)
-(THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 t2))) k0 H6)))) H3))
-H2)))))))) y u2 H0))) H))))).
-
-theorem iso_flats_lref_bind_false:
- \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall
-(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind
-b) v t)) \to (\forall (P: Prop).P)))))))
-\def
- \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda
-(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads
-(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))
-(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let
-H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in
-(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P
-(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let
-H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef x) H1) in
-(False_ind P H2)))) H0))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda
-(_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall
-(P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1
-(TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def
-(iso_gen_head (Flat f) t0 (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)
-H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda (v2: T).(\lambda (t2:
-T).(eq T (THead (Bind b) v t) (THead (Flat f) v2 t2)))) P (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f)
-x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat f) x0 x1) H2) in (False_ind P H3))))) H1))))))))
-vs)))))).
-
-theorem iso_flats_flat_bind_false:
- \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall
-(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads
-(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P:
-Prop).P)))))))))
-\def
- \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda
-(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind
-(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2))
-(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead
-(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def
-(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in
-(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t)
-(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1:
-(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T
-(THead (Bind b) v t) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1:
-TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))
-(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead
-(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v
-t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads
-(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1
-\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead
-(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3
-\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1))))))))
-vs)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/iso/fwd.ma".
-
-theorem iso_refl:
- \forall (t: T).(iso t t)
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n:
-nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k:
-K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_:
-(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t).
-
-theorem iso_trans:
- \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
-t3) \to (iso t1 t3)))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
-(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
-(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in
-(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3)
-(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x)
-(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1)))))))
-(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso
-(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in
-(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3)
-(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x)
-(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1)))))))
-(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def
-(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda
-(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5)
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0
-x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t))
-(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/leq/props.ma".
-
-theorem asucc_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
-(asucc g a1) (asucc g a2)))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
-a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
-nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
-(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k)
-(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O
-(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow
-(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq
-A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n:
-nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g
-(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S
-h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k)
-(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind
-A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O
-(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq
-A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k)
-(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k))))
-(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k)
-H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g
-(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3:
-nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
-\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
-g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
-(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1)
-n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g
-(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a:
-A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3)
-n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2)
-k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort
-O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k))
-(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1))
-(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g
-(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g
-n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort
-O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A
-(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda
-(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to
-((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g
-(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow
-(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h)
-\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O
-\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))
-(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
-k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
-\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
-\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1
-(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A
-(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k))
-(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g
-(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O
-n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort
-(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k))
-(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda
-(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
-h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k))
-\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
-\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
-n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
-with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
-n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
-(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
-(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
-g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4
-n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a
-(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k))
-(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A
-(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g
-(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S
-h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
-(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k)
-(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda
-(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g
-(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
-(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
-a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
-
-theorem asucc_inj:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
-g a2)) \to (leq g a1 a2))))
-\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
-(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
-(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
-(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
-n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
-(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
-n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
-(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
-(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1
-g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind
-nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
-(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort
-h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
-x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2))
-(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3
-O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g
-(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in
-(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a:
-A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S
-x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2))
-a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
-(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3:
-nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2)))
-\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g
-(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O
-(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda
-(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g
-(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1:
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
-x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
-\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
-n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1
-x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1
-(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
-n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8
-n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
-(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0)
-(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10
-(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in
-(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0))
-(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g
-(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq
-g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda
-(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to
-((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort
-n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))
-(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
-n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
-\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3
-n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g
-(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus
-g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4:
-nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7)
-in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3
-n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def
-(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
-(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
-(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
-x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
-(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4:
-nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
-n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
-g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
-n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
-n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
-n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def
-(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind
-nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
-(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2
-n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0:
-nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g
-(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4
-n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
-\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
-n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1
-x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1
-(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0)
-x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A
-(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10
-\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
-(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
-(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2))
-a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2))
-in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1
-H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n
-n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda
-(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
-a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
-a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
-a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
-(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
-n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
-(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
-n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
-g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
-(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc
-g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g
-n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g
-(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda
-(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g
-(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1
-x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
-(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g
-(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_:
-(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a)))
-\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0)
-a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g
-(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
-n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g
-(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0)
-a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
-a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in
-(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2)
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a
-(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0))
-(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A
-(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A
-(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc
-g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1
-x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8)))))))
-H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2:
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0:
-A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to
-(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g
-(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g
-(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0))
-(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda
-(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def
-(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x
-in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda
-(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda
-(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0)
-(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a
-x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next
-g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
-(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1:
-nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0)))
-\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g
-(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a
-(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda
-(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq
-g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
-(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g
-a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1)
-n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g
-(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda
-(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g
-(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g
-(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3
-(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5:
-A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g
-(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g
-a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g
-a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8
-\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
-with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3
-(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
-((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n)
-\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h)
-\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0
-a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g
-a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def
-(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9)
-in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in
-(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2))))))
-a1)).
-
-theorem leq_asucc:
- \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
-a0)))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
-A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
-A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
-(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
-(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
-(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
-(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
-(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
-g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
-(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
-a)).
-
-theorem leq_ahead_asucc_false:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
-(asucc g a1)) \to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
-A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
-(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
-(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
-\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
-nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
-(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
-(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1
-g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind
-A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_:
-A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
-(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda
-(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A
-(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1:
-nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O
-\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to
-P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let
-H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1
-\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S
-n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_:
-(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H))))))
-(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g
-a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall
-(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P:
-Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2)
-(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g
-(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind
-A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_:
-A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
-(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
-x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g
-a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e
-in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc
-(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0
-with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) |
-(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead
-_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda
-(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3))
-H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g
-(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6)))))))
-H2)))))))))) a1)).
-
-theorem leq_asucc_false:
- \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
-Prop).P)))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
-a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
-(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
-\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
-(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
-n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
-(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O
-(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda
-(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda
-(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
-(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A
-(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e:
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow
-n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
-with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r
-nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g
-(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7
-n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
-(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O
-n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S
-x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1:
-nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda
-(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next
-g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda
-(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g
-n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda
-(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
-(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S
-n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e
-in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _
-_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1)
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def
-(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g
-(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda
-(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2)))
-H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda
-(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1)
-n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g
-(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2:
-nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H)))))
-(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P:
-Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to
-(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead
-a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1)
-(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
-(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1)
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0
-x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1)
-(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _)
-\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1)
-H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2:
-A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0
-(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6)))))))
-H2))))))))) a)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/aplus/defs.ma".
-
-inductive leq (g: G): A \to (A \to Prop) \def
-| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
-(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
-h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
-| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3:
-A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2
-a4))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/leq/defs.ma".
-
-theorem leq_gen_sort1:
- \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
-g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
-(ASort h2 n2))))))))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda
-(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort
-h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A
-a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g
-(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat
-nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a
-k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0:
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
-k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal
-A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
-n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1
-n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0
-h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0
-n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
-nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3
-n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort
-n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0)))))
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2)
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3
-n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
-(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0
-H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
-(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k)
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
-nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2
-n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def
-(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k)
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0)))
-H))))).
-
-theorem leq_gen_head1:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
-(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1
-a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
-g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
-a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
-(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq
-A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1)
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
-(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
-(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2)
-(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
-(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A
-(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda
-(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4
-a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
-a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
-A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2
-a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6
-a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4)
-(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6)
-\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0
-a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6:
-A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6:
-A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
-A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda
-(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12
-\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A
-A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda
-(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5)
-(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5)))))))))
-H6))))))))))) y a H0))) H))))).
-
-theorem leq_gen_sort2:
- \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
-g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1)
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
-(ASort h2 n2))))))))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda
-(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
-(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq
-A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind
-g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat
-nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus
-g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0:
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
-k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal
-A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
-n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1
-n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2
-h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0
-n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
-nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3
-n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort
-h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0)))))
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0)
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1
-n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
-(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2
-H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
-(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
-nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2
-n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def
-(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
-(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0)))
-H))))).
-
-theorem leq_gen_head2:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a
-(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3
-a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
-g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
-a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
-(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq
-A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2)
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
-(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
-(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1)
-(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
-(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A
-(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda
-(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4
-a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
-a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
-A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7
-a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6
-a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5)
-(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6)
-\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3
-a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6:
-A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6:
-A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
-A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda
-(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12
-\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A
-A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda
-(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4)
-(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4)))))))))
-H6))))))))))) a y H0))) H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/leq/fwd.ma".
-
-include "LambdaDelta-1/aplus/props.ma".
-
-theorem ahead_inj_snd:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
-(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
-(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def
-(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A
-(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda
-(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4)
-(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda
-(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead
-a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in
-A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _)
-\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3)
-in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a:
-A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a:
-A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
-
-theorem leq_refl:
- \forall (g: G).(\forall (a: A).(leq g a a))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
-(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
-(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
-a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
-H0))))) a)).
-
-theorem leq_eq:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
-a2))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
-a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
-
-theorem leq_sym:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
-a2 a1))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
-nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
-(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
-(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
-A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
-H1 a6 a5 H3))))))))) a1 a2 H)))).
-
-theorem leq_trans:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
-(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
-a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
-(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2
-\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3
-n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3
-(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1:
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus
-g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def
-(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1
-x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1
-n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g
-(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r
-nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n)
-(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k)
-x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus
-g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7
-H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2)
-(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus
-(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g
-(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1
-x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g
-(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3:
-A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5:
-A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6:
-A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7)
-\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6)
-a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in
-(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8:
-(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0
-(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead
-a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9)))))))
-H5))))))))))))) a1 a2 H)))).
-
-theorem leq_ahead_false_1:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
-\to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
-A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
-n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
-(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
-(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0)
-a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4:
-A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
-(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O
-n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee:
-A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
-True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P
-H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1
-n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0)
-a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2
-(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1)
-n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
-(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort
-(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
-(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H:
-((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
-Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
-a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
-g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def
-(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in
-(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3)))
-(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda
-(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
-x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5)
-in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
-(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3]))
-(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def
-(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def
-(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0
-H10 P))))) H6))))))) H2)))))))))) a1)).
-
-theorem leq_ahead_false_2:
- \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
-\to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
-A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
-n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
-(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
-a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O
-n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
-(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
-x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0)
-(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
-(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P
-H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1
-n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1)
-n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0)
-(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
-A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
-(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1)
-n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
-a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort
-(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
-(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_:
-((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P:
-Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead
-a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq
-g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def
-(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in
-(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_:
-A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4:
-A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
-A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0)
-x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5)
-in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
-(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3]))
-(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def
-(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let
-H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9
-P))))) H6))))))) H2)))))))))) a2)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tlist/defs.ma".
-
-include "LambdaDelta-1/s/defs.ma".
-
-definition lref_map:
- ((nat \to nat)) \to (nat \to (T \to T))
-\def
- let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t
-with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u
-t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in
-lref_map.
-
-definition lift:
- nat \to (nat \to (T \to T))
-\def
- \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x:
-nat).(plus x h)) i t))).
-
-definition lifts:
- nat \to (nat \to (TList \to TList))
-\def
- let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with
-[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts
-h d ts0))]) in lifts.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift/defs.ma".
-
-theorem lift_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
-n)) (TSort n))))
-\def
- \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort
-n)))).
-
-theorem lift_lref_lt:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T
-(lift h d (TLRef n)) (TLRef n)))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
-d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T
-(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
-
-theorem lift_lref_ge:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
-(lift h d (TLRef n)) (TLRef (plus n h))))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
-n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h))))
-(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false
-(le_bge d n H)))))).
-
-theorem lift_head:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d)
-t)))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
-
-theorem lift_bind:
- \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u)
-(lift h (S d) t)))))))
-\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
-
-theorem lift_flat:
- \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u)
-(lift h d t)))))))
-\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
-
-theorem lift_gen_sort:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T
-(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind
-(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))
-(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort
-n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H:
-(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort
-n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
-(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let
-H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
-(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0)
-H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d
-n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
-(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
-(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef
-(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda
-(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq
-T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d
-t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d
-(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
-(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1))
-(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k
-t0 t1) (TSort n)) H3))))))))) t)))).
-
-theorem lift_gen_lref:
- \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
-(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le
-(plus d h) i) (eq T t (TLRef (minus i h)))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h:
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
-(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
-h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda
-(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def
-(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort
-n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i)))
-(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1))))))))
-(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
-nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or
-(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T
-(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
-T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n)
-(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
-n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef
-n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0
-h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus
-d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n)
-(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d
-n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef
-i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h)
-(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le
-(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n
-(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n
-h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0)))))
-(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
-(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h)
-(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h))
-(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i
-H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d:
-nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to
-(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0
-(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d:
-nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to
-(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1
-(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
-nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def
-(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1
-(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
-H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d)
-t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i)))
-(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h)))))
-H3)))))))))))) t).
-
-theorem lift_gen_lref_lt:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall
-(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n
-d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x
-\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d)
-(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq
-T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind
-(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n
-d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2:
-(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d
-h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus
-d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
-(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq
-T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S
-n) d h H))) t H4))) H2)) H1)))))))).
-
-theorem lift_gen_lref_false:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
-(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall
-(P: Prop).P)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
-n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T
-(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d
-h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n)))
-(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land
-(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P
-(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H
-H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n
-h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
-(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false
-(plus d h) n P H4 H0))) H3)) H2)))))))))).
-
-theorem lift_gen_lref_ge:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
-(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
-n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
-t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in
-(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus
-d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n))
-(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind
-(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda
-(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r
-T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq
-T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d
-(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2))
-(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
-h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
-h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
-(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n
-h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus
-(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
-
-theorem lift_gen_head:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z)))))))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
-(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
-(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n:
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
-(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda
-(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1
-\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef
-n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
-(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0:
-(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
-(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def
-(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def
-(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H
-(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead
-k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0:
-T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
-(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1:
-T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
-(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0
-t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2:
-T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
-(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0
-(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2]))
-(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2)
-\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1))
-H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let
-H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k
-H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z:
-T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
-k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let
-H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0:
-nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift
-h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k
-d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
-(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d)
-z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
-t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10
-(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
-t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9
-(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z))))))
-(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d)
-z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0))
-(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4))
-H3))))))))))) x)))).
-
-theorem lift_gen_bind:
- \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T
-T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (S d) z)))))))))))
-\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d
-x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
-h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
-(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
-(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d)
-x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
-(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d)
-z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro
-T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind
-b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
-z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d
-x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
-
-theorem lift_gen_flat:
- \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T
-T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h d z)))))))))))
-\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d
-x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
-h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
-(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
-(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T
-T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0:
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1)
-(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T
-(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
-T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d
-x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
-(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
-(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1))
-(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x
-H1)))))) H0))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift/fwd.ma".
-
-include "LambdaDelta-1/s/props.ma".
-
-theorem thead_x_lift_y_y:
- \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
-(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
-\def
- \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v:
-T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0)
-\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n)))
-(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d
-(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n:
-nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
-(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def
-(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
-(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_:
-((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
-h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0:
-((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
-h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0
-t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
-(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _)
-\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
-H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map
-(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort
-n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
-with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
-\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
-lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat
-\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
-\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
-lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow
-(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T
-\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f
-i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f
-(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec
-lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with
-[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
-d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
-\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
-lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2)
-\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
-H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def
-(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall
-(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0:
-Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0
-t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0
-d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P))))))
-H3)) H2)))))))))))) t)).
-
-theorem lift_r:
- \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0)
-t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda
-(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n))
-(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef
-n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H)))
-(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T
-t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
-(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k:
-K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
-t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
-t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
-t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k
-(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d)))
-(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
-
-theorem lift_lref_gt:
- \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
-(pred n))) (TLRef n))))
-\def
- \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef
-(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
-(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
-(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
-(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n)))
-(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
-(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
-(S_pred n d H))))))).
-
-theorem lifts_tapp:
- \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq
-TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs:
-TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp
-(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil))
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp
-t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d
-t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1)
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0
-v)) H)))) vs)))).
-
-theorem lift_inj:
- \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
-(lift h d x) (lift h d t)) \to (eq T x t)))))
-\def
- \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
-t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
-(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
-(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
-H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
-T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
-(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
-n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
-d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
-h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
-(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
-t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
-T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
-(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1:
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1))
-\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
-nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
-t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
-(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall
-(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0
-t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
-(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
-(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1
-(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
-(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
-h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
-T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
-(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b))
-(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t)
-(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t:
-T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T
-(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda
-(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d
-t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0))
-(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0))
-(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t)
-(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y:
-T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq
-T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda
-(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0)
-(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T
-(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0
-x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3))))))
-(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
-
-theorem lift_gen_lift:
- \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
-nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
-t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
-t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
-nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
-((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
-T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
-t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
-(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
-d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
-x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
-(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
-(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
-(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
-(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
-(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
-(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
-d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
-(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
-(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
-(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
-(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
-h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
-(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
-d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
-(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
-(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
-h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
-(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
-(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
-T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
-(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
-n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
-(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
-(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
-(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
-(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
-(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
-(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
-(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
-T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
-(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
-(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
-n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
-t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
-h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
-(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
-(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2)))
-(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq
-T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
-(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus
-n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2)
-h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4)
-(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda
-(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2
-d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus
-n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus
-(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans
-h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T
-(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T
-t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))
-(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1
-d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n
-h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0)
-(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2)
-h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T
-(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2)))
-(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2
-n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2
-h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2)
-(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2)))))
-(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t:
-T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus
-n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat
-plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus
-(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2))
-(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2)))
-(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus
-(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4)
-(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n
-(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n
-h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
-(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift
-h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift
-h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda
-(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2:
-nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
-t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
-t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x:
-T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2:
-nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t
-t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
-(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T
-x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2
-t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t
-t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
-(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3
-(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1))
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2
-h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2
-(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
-(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda
-(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
-(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind
-b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
-(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T
-(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
-d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1
-t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2))))
-(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
-t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
-(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
-T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
-x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1
-d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0)
-(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let
-H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1)
-t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2:
-T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2)
-t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift
-h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift
-h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
-x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S
-d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift
-h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
-h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda
-(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift
-h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
-h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead
-(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2:
-T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2
-t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2)
-(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
-(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift
-h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1
-d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
-(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
-t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)))
-(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1
-H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8))))
-(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1
-(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T
-(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def
-(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift
-h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0))
-(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z:
-T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2:
-T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0)
-(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x
-(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2
-h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1)
-x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda
-(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t
-t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1
-t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq
-T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead
-(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0
-(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift
-h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2
-x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2
-d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3:
-T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3:
-T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2:
-T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2)))
-(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1
-t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2
-t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda
-(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1
-d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2
-d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
-T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3)))
-(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2
-t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2)
-(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f)
-(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3)
-(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2:
-T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T
-(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f)
-x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2
-x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2)
-(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2
-d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0
-H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2
-H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus
-d2 h1) H4))))) k H2))))))))))))) t1).
-
-theorem lifts_inj:
- \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
-nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
-\def
- \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts:
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
-d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t:
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts
-h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_:
-nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda
-(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d:
-nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t)
-(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match
-ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True |
-(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in
-(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t:
-T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h:
-nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq
-TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1:
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t
-t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d
-t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0))
-(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with
-[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in
-(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda
-(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList
-(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t
-t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList
-(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d
-t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList
-return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f:
-((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow
-(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda
-(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d
-t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def
-(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda
-(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat)
-(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil |
-(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in
-lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h
-d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift
-h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0)
-(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H
-t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
-
-theorem lift_free:
- \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
-nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
-(lift h d t)) (lift (plus k h) d t))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
-nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
-(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n:
-nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
-nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T
-(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort
-n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
-(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0))
-(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k
-h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n))
-(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k:
-nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d
-h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef
-n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T
-(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef
-n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
-(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
-(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus
-k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1
-H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d
-n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift
-(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda
-(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n
-(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal
-nat T TLRef (plus (plus n h) k) (plus n (plus k h))
-(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
-(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
-(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1
-(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
-(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0:
-nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
-(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda
-(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d:
-nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e
-(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda
-(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d
-h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k
-d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0
-t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift
-h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0
-t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d)
-t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k
-e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h
-d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift
-(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d)
-(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le
-k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
-(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e
-(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift
-h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h
-d))))))))))))) t).
-
-theorem lift_d:
- \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
-nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t))
-(lift k e (lift h d t))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
-nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
-d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda
-(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_:
-(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0)
-(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
-T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0:
-T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
-T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k
-e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n))
-(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e))))))))
-(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
-nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h
-(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda
-(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n)
-(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef
-n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d
-(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k
-e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
-(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift
-h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n))
-(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k
-d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e
-n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d)
-t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0:
-nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n)))))
-(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d
-(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda
-(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n)
-(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef
-(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
-(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
-(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
-(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1:
-(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0
-(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda
-(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef
-(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0))
-(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat
-(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k
-e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0)))
-(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus
-n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n
-k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e
-H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h:
-nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq
-T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d
-t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0:
-nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0
-d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h:
-nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le
-e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
-T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
-(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
-k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d
-(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1))
-(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h
-(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead
-k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda
-(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
-k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda
-(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift
-k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h
-(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e
-t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e)
-t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1)
-(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0
-d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k
-(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1))
-(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0
-(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0
-d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift/fwd.ma".
-
-include "LambdaDelta-1/tlt/props.ma".
-
-theorem lift_weight_map:
- \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
-nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
-(weight_map f (lift h d t)) (weight_map f t))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d:
-nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
-(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0)))))))
-(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
-nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m)
-O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n:
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
-nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m)
-O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f
-(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat
-(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0)))
-(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq
-nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda
-(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0))
-(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda
-(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d:
-nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
-(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f
-t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d:
-nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
-(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f
-t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
-nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m)
-O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
-t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead
-(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat
-(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda
-(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift
-h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d)
-t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map
-(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f
-(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with
-[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S
-(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0)
-(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0)
-(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n:
-nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S
-(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))))
-(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n:
-nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0)
-(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus
-(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))
-(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d)
-(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d)
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
-n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda
-(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda
-(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3))))
-(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r
-nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map
-f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O)
-t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
-(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2
-nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map
-(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat
-(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h
-(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat
-(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd
-f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le
-d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x
-H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1)
-(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus
-(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus
-(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f
-t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f
-(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd
-f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1))))
-(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m:
-nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
-n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x:
-nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d
-m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h
-d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s
-(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f
-(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d
-t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1))
-(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0)
-(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1)))
-(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
-k)))))))))) t).
-
-theorem lift_weight:
- \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d
-t)) (weight t))))
-\def
- \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d
-(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat
-O)))))).
-
-theorem lift_weight_add:
- \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d:
-nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to
-(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
-(weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
-\def
- \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h:
-nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat
-(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))
-\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
-t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m:
-nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d)
-w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
-m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n))))))))))))
-(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m
-d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1:
-((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d
-(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d
-(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n))
-(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef
-n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d
-H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0:
-T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
-(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f
-(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda
-(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f
-(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h))
-(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift
-h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda
-(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to
-(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d
-m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0))
-(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0:
-((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall
-(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f
-m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g
-(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift
-(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m:
-nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d)
-w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
-m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
-t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b:
-B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1))
-(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead
-(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h)
-(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b)
-(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind
-(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus
-(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d
-t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h
-d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S
-(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
-t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h)
-d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h)
-(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0))
-(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
-t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
-(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift
-(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map
-f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S
-(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S
-(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2
-H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S
-(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S
-d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
-(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d
-t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m
-O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift
-(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat
-nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq
-nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g
-H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S
-m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat
-m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g
-(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda
-(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r
-nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d
-t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6))))
-H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
-n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x:
-nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0)))
-n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus
-(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
-t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g
-(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map
-(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O)
-(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O)
-(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))
-(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat
-O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m
-H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda
-(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0)))
-(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x:
-nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6))))
-H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
-n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S
-x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g
-n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat
-S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
-t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S
-h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0))
-(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1))
-(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S
-d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S
-d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
-(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda
-(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n)
-(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0:
-nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda
-(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O
-m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda
-(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n)
-(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m:
-nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
-n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x:
-nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5))))
-(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head
-(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind
-b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0)
-(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2)
-(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead
-(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2:
-T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0)
-d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d
-t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0))
-(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f
-(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1))
-(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3)))
-(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d))
-(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
-k))))))))))))) t)).
-
-theorem lift_weight_add_O:
- \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to
-nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h)
-O t))))))
-\def
- \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to
-nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m:
-nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m)))))
-(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal
-nat (f m)))))))).
-
-theorem lift_tlt_dx:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(tlt t (THead k u (lift h d t)))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight
-(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t)
-(lift_weight t h d)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift/defs.ma".
-
-definition trans:
- PList \to (nat \to nat)
-\def
- let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i:
-nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let
-j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false
-\Rightarrow (plus j h)]))])) in trans.
-
-definition lift1:
- PList \to (T \to T)
-\def
- let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds
-with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0
-t))])) in lift1.
-
-definition lifts1:
- PList \to (TList \to TList)
-\def
- let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with
-[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t)
-(lifts1 hds ts0))]) in lifts1.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift1/defs.ma".
-
-include "LambdaDelta-1/lift/fwd.ma".
-
-theorem lift1_sort:
- \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
-\def
- \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T
-(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0:
-nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p
-(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0
-n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
-
-theorem lift1_lref:
- \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef
-(trans hds i))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T
-(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T
-(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
-(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p
-i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq
-T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow
-(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T
-(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false
-\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
-
-theorem lift1_bind:
- \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
-(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss
-hds) t))))))
-\def
- \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
-(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
-(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal
-T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
-PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
-(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda
-(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p)
-t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p
-u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n
-n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0
-(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t)))))
-(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1
-(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t)))
-(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u
-t)) (H u t)))))))) hds)).
-
-theorem lift1_flat:
- \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
-(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds
-t))))))
-\def
- \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
-(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f)
-(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T
-(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
-PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
-(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u:
-T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t))
-(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u))
-(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p
-u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift
-n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f)
-(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f)
-(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1
-p (THead (Flat f) u t)) (H u t)))))))) hds)).
-
-theorem lift1_cons_tail:
- \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq
-T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
-\def
- \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds:
-PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t)
-(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1
-(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d
-t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d
-t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p
-h d) t) H))))) hds)))).
-
-theorem lifts1_flat:
- \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts:
-TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds
-ts) (lift1 hds t))))))
-\def
- \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts:
-TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0
-t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1
-hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds
-(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds
-t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f)
-t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads
-(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f)
-(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1
-hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1)
-(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat
-f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H)
-(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0
-(THeads (Flat f) t1 t)))))) ts)))).
-
-theorem lifts1_nil:
- \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
-\def
- \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t)
-t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H:
-(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq
-TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1
-PNil t0) H)))) ts).
-
-theorem lifts1_cons:
- \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts:
-TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts:
-TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t)
-(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda
-(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d
-(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1:
-TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1
-hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d
-(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0)
-H)))) ts)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift/props.ma".
-
-include "LambdaDelta-1/drop1/defs.ma".
-
-theorem lift1_lift1:
- \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1
-(lift1 is2 t)) (lift1 (papp is1 is2) t))))
-\def
- \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2:
-PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2)
-t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t))))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H:
-((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1
-(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat
-T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal
-nat n) (refl_equal nat n0) (H is2 t)))))))) is1).
-
-theorem lift1_xhg:
- \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t))
-(lift (S O) O (lift1 hds t))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T
-(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t:
-T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d:
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p)
-(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T
-(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S
-O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n:
-nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d
-(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda
-(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift
-(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1
-p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S
-d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
-
-theorem lifts1_xhg:
- \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts
-(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
-\def
- \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq
-TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t))))
-(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq
-TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds
-t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList
-(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1
-hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O
-(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1
-hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds
-t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
-(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds)
-(lift (S O) O t)) (lift1_xhg hds t))))) ts)).
-
-theorem lift1_free:
- \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds
-(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i:
-nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O
-(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T
-(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0:
-PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0
-(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i)
-t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0
-i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift
-(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) |
-false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0
-i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans
-hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt
-(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i))
-O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans
-hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with
-[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) |
-false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind
-(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d
-(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with
-[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
-h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))))
-(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S
-(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h
-n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0
-i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t))))
-(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i)))
-(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0
-i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t))))
-(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans
-hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S
-(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t)))
-(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans
-hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S
-(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d
-(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0:
-(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans
-hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S
-(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus
-h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i)
-t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t))))
-(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O
-(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans
-hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1
-(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i)))
-(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S
-(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0
-i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i)))
-(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda
-(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d
-(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i))
-(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans
-hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t))))))))
-hds).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/A/defs.ma".
-
-definition lweight:
- A \to nat
-\def
- let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow
-O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in
-lweight.
-
-definition llt:
- A \to (A \to Prop)
-\def
- \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/llt/defs.ma".
-
-include "LambdaDelta-1/leq/defs.ma".
-
-theorem lweight_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
-(lweight a1) (lweight a2)))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight
-a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
-nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g
-(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3:
-A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight
-a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda
-(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight
-a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
-(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2
-H)))).
-
-theorem llt_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
-(a3: A).((llt a1 a3) \to (llt a2 a3))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
-a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1
-\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0
-(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
-
-theorem llt_trans:
- \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2
-a3) \to (llt a1 a3)))))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight
-a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans
-(lweight a1) (lweight a2) (lweight a3) H H0))))).
-
-theorem llt_head_sx:
- \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1)
-(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))).
-
-theorem llt_head_dx:
- \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1)
-(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))).
-
-theorem llt_wf__q_ind:
- \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to
-Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0
-a))))) P n))) \to (\forall (a: A).(P a)))
-\def
- let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
-A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
-n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight
-a)))))).
-
-theorem llt_wf_ind:
- \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1
-a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
-\def
- let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
-A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
-Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
-(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind
-(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0:
-A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
-\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
-(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
-(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P
-a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt
-(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight
-a1))))))))))))) a)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/G/defs.ma".
-
-definition next_plus:
- G \to (nat \to (nat \to nat))
-\def
- let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O
-\Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/next_plus/defs.ma".
-
-theorem next_plus_assoc:
- \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq
-nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0:
-nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2)
-(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus
-g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat
-(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda
-(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus
-g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda
-(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1))))
-(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0))
-(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n
-n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1))
-(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0))
-n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g
-(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0)
-(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))).
-
-theorem next_plus_next:
- \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g
-(next g n) h) (next g (next_plus g n h)))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus
-g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n
-h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n
-(S O)) h) (next_plus_assoc g n (S O) h)))).
-
-theorem next_plus_lt:
- \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next
-g n) h))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0:
-nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n))
-(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next
-g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0))
-n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next
-g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next
-g n0) n)) (next_plus_next g (next g n0) n))))) h)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/nf2/fwd.ma".
-
-include "LambdaDelta-1/arity/subst0.ma".
-
-theorem arity_nf2_inv_all:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat
-(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
-A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
-(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda
-(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w:
-T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
-c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort
-n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort
-n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n
-(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u)
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0:
-nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0:
-nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef
-i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda
-(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
-(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2
-TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads
-(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
-ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
-i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
-A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u
-(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef
-i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda
-(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i)
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
-(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
-(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
-i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
-(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
-(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
-i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
-(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0
-(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0
-(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind
-b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
-(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2
-(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead
-c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u
-t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0
-(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
-(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq
-B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0
-a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0
-H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr)
-u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
-w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u
-t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g
-(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u
-t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False return (\lambda
-(_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
-(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
-(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void
-Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8:
-(nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0
-(Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda
-(v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda
-(u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
-Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda
-(H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1:
-T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T
-(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda
-(u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
-Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11
-(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u
-(lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O
-x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
-T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3
-H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
-(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T
-T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
-(Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to
-(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst)
-w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList
-nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind
-Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind
-Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u
-t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2
-(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0:
-T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u
-t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
-T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2
-(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda
-(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
-Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind
-Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
-(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
-t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def
-(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0)
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList
-nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0)
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def
-(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda
-(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
-w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9:
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w:
-T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
-c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
-T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
-t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
-T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10:
-(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_:
-(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1:
-T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in
-(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2
-(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1)
-(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
-(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
-(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1)
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
-(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind
-Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind
-Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat
-Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10))))))))
-H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind
-nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x:
-nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda
-(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def
-(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x)
-H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl)
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def
-(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12))
-in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq
-g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead
-(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
-(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
-(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x))
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
-x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0
-x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
-\Rightarrow False])) I (AHead x0 x1) H16) in (False_ind (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead
-(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
-(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
-(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x))
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))))) H17))))))) H13))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
-(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0:
-TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0
-(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef
-x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl)
-u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T
-t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0
-(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda
-(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
-Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
-t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
-(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
-(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads
-(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda
-(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u
-(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11)
-H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda
-(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda
-(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda
-(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0
-u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
-Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u
-t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
-(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0
-t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
-Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
-nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda
-(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
-u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
-Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind
-Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind
-Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
-u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead
-(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
-u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))))
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
-Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u:
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
-c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8))
-t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T
-(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u:
-T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2
-c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))))
-(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda
-(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
-T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort
-x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
-(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
-(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x
-(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
-(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
-T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
-c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
-(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
-ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda
-(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0
-(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef
-x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
-u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads
-(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
-c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl)
-x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
-(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
-(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
-x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
-i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0
-H6)))))) H5)) H4))))))))))) c t a H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/nf2/defs.ma".
-
-include "LambdaDelta-1/pr2/clen.ma".
-
-include "LambdaDelta-1/pr2/fwd.ma".
-
-include "LambdaDelta-1/pr0/dec.ma".
-
-include "LambdaDelta-1/C/props.ma".
-
-theorem nf2_dec:
- \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq
-T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
-\def
- \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall
-(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1
-t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda
-(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in
-(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))
-(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to
-(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2
-(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2
-H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2:
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))
-(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to
-(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2:
-T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T
-t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))
-(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x
-H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or
-(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1
-t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H
-t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0)
-t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1:
-((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0:
-K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2
-T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0:
-B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1
-t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def
-(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v:
-T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O)
-(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2)
-\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda
-(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq
-T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O)
-(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2
-(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
-(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O)
-(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def
-H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t
-c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind
-Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2)
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1
-t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t
-c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2
-(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
-(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1
-t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0)
-x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0
-(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in
-(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0)
-(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt
-(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym
-(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t
-(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5))))
-(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1
-(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1
-(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda
-(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T
-t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall
-(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T
-(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O)
-(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
-(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda
-(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1
-\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let
-H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda
-(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O)
-(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift
-(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x)
-t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
-Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0:
-T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr)
-(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda
-(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2)
-(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10:
-(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0
-t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3)))
-(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1:
-T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x
-x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2))
-H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O)
-(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0))
-(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen
-c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x)
-t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4)))
-H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2)
-\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda
-(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def
-(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
-(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr)))
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
-(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
-(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0))
-(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
-K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
-T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
-(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
-(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee:
-K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow
-(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
-Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow
-False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))
-(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2:
-T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def
-(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
-(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr)))
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
-(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
-(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0))
-(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
-K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
-T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
-(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
-(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee:
-K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow
-(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
-Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow
-False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))
-b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0)
-t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda
-(t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def
-(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2
-c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0:
-T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2)
-(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_:
-T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
-T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f)
-(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen
-c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f)
-(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0
-t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))))
-k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2:
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))
-(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x)
-\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall
-(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t
-c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1
-x H3 k t)))))) H1)) H0)))))))) c).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr2/defs.ma".
-
-definition nf2:
- C \to (T \to Prop)
-\def
- \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1
-t2)))).
-
-definition nfs2:
- C \to (TList \to Prop)
-\def
- let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil
-\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))])
-in nfs2.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/nf2/defs.ma".
-
-include "LambdaDelta-1/pr2/clen.ma".
-
-include "LambdaDelta-1/subst0/dec.ma".
-
-include "LambdaDelta-1/T/props.ma".
-
-theorem nf2_gen_lref:
- \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2
-c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P:
-Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0
-(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef
-i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
-
-theorem nf2_gen_abst:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u
-t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
-T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t)
-t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2:
-T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2:
-T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
-(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst)
-u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2
-H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u
-t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u)
-t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t
-t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
-_ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H
-(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in
-H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind
-Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0))
-(refl_equal T t) t2 H1))))))))).
-
-theorem nf2_gen_cast:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u
-t)) \to (\forall (P: Prop).P))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
-(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t
-(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
-
-theorem nf2_gen_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
-(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
-((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2)
-\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P:
-Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind
-Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead
-(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in
-(False_ind P H0))))))).
-
-theorem nf2_gen_flat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
-(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
-((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f)
-u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall
-(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c
-u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t)
-(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1)))
-(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
-(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
-(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
-
-theorem nf2_gen__nf2_gen_aux:
- \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
-(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u:
-T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
-(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
-nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort
-n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
-d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n:
-nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u
-(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind
-T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
-(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall
-(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
-(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u:
-T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to
-(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1:
-(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
-t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
-_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
-(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1]))
-(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let
-H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat)))
-(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort
-n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
-| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0
-(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0:
-nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat)
-(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) |
-(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i |
-false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map
-f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus
-x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map
-(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort
-n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
-with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
-\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f:
-((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
-\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1)
-\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
-t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7
-\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0))
-H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t
-t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
-(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8
-P)))))) H3)) H2))))))))))) x)).
-
-theorem nf2_gen_abbr:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u
-t)) \to (\forall (P: Prop).P))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
-T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t)
-t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x
-in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t
-(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift
-(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O
-x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O
-x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
-_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S
-O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind
-Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u)
-t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda
-(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in
-(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O)
-O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c
-(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H
-(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c
-(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x
-(pr0_refl x) u))) P))) H1))) H0))))))).
-
-theorem nf2_gen_void:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
-(lift (S O) O t))) \to (\forall (P: Prop).P))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
-T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
-Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux
-Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t
-(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u)))
-P))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/nf2/pr3.ma".
-
-include "LambdaDelta-1/pr3/fwd.ma".
-
-include "LambdaDelta-1/iso/props.ma".
-
-theorem nf2_iso_appls_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
-TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso
-(THeads (Flat Appl) vs (TLRef i)) u))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
-(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads
-(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u))))
-(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def
-(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda
-(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda
-(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t:
-T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat
-Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i))
-u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat
-Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl)
-t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
-T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
-Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2:
-T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
-i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u
-(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat
-Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0
-x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0
-(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0
-(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda
-(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
-T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1
-t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u)
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
-(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda
-(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0
-x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
-u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in
-(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
-(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
-Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2:
-T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
-u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
-Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
-x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift
-(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1
-x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0
-(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0
-H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
-u))))))))))))))) H3)) H2))))))) vs)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/nf2/props.ma".
-
-include "LambdaDelta-1/drop1/fwd.ma".
-
-theorem nf2_lift1:
- \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1
-hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
-\def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
-(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p
-t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c
-e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in
-(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c:
-C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p
-t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p)
-c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0)
-in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda
-(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x:
-C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x
-(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/nf2/defs.ma".
-
-include "LambdaDelta-1/pr3/pr3.ma".
-
-theorem nf2_pr3_unfold:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c
-t1) \to (eq T t1 t2)))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t
-t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t
-(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0)
-\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def
-(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def
-(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T
-t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
-
-theorem nf2_pr3_confluence:
- \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2)
-\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2:
-T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t
-t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0))
-(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3:
-(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1
-x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1
-H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y)
-in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2
-(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0:
-T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2
-t1 H1))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/nf2/defs.ma".
-
-include "LambdaDelta-1/pr2/fwd.ma".
-
-theorem nf2_sort:
- \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
-\def
- \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort
-n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal
-T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
-
-theorem nf2_csort_lref:
- \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i)))
-\def
- \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort
-n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq
-T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n)
-(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S
-i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r
-T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2
-H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort
-n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift
-(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort
-n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift
-(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2
-(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T
-(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i)
-(lift (S i) O x1))) t2 H3))))) H1)) H0))))).
-
-theorem nf2_abst:
- \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v:
-T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind
-Abst) u t))))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
-\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda
-(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t
-t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t)
-t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead
-(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2
-(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5:
-((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t
-x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead
-(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t
-x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3))))))
-H2)))))))))).
-
-theorem nf2_abst_shift:
- \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c
-(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
-\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2
-(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda
-(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2
-H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
-u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda
-(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2
-c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
-b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T
-(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst)
-u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2
-H3)))))) H2)))))))).
-
-theorem nfs2_tapp:
- \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t))
-\to (land (nfs2 c ts) (nf2 c t)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0:
-TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H:
-(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True
-(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I
-H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c
-(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c
-t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c
-(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2:
-(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let
-H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c
-t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj
-(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5)
-H6))) H4))))) H1)))))) ts))).
-
-theorem nf2_appls_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
-TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
-(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads
-(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda
-(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0
-(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in
-(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat
-Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c
-t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def
-(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
-t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
-(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
-(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t
-x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T
-(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1:
-T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0
-(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i))
-(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
-i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1:
-T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1
-(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2
-H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
-t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst)
-x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2
-c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
-u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind
-(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T
-(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead
-(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2
-x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda
-(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0
-x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
-x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil
-(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3:
-TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T
-(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead
-(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2
-x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef
-i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i))
-(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1
-(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat
-Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2
-x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T
-T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0:
-B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10:
-(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
-x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead
-(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl)
-t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1:
-TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat
-Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t
-(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl)
-(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil
-(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead
-(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
-(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat
-Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
-x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c
-(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef
-i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat
-Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
-x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef
-i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i))
-(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1
-(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat
-Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5
-(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2
-H10))))))))))))) H7)) H6))))))) H2)))))) vs)))).
-
-theorem nf2_appl_lref:
- \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c
-(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i:
-nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0
-(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))).
-
-theorem nf2_lref_abst:
- \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c
-(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c
-(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2
-(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d
-(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O
-u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T
-(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2
-H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c
-(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift
-(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c
-(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift
-(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i)
-O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t))
-(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c
-c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H
-(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst)
-u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort
-_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind
-Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1)
-H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2))
-H1)))))))).
-
-theorem nf2_lift:
- \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h:
-nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
-\def
- \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2)
-\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i:
-nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c
-(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind
-T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3))
-(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i
-x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq
-T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x
-(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq
-T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3))))
-H2)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr1/defs.ma".
-
-definition pc1:
- T \to (T \to Prop)
-\def
- \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda
-(t: T).(pr1 t2 t)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc1/defs.ma".
-
-include "LambdaDelta-1/pr1/pr1.ma".
-
-theorem pc1_pr0_r:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T
-(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H)
-(pr1_refl t2)))).
-
-theorem pc1_pr0_x:
- \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T
-(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1)
-(pr1_pr0 t2 t1 H)))).
-
-theorem pc1_refl:
- \forall (t: T).(pc1 t t)
-\def
- \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0:
-T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
-
-theorem pc1_pr0_u:
- \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2
-t3) \to (pc1 t1 t3)))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3:
-T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t:
-T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x:
-T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda
-(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2)
-H3)))) H1)))))).
-
-theorem pc1_s:
- \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in
-(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2
-t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2
-x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2
-H1)))) H0)))).
-
-theorem pc1_head_1:
- \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall
-(k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t:
-T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0))
-(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda
-(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T
-(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2
-t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k)))))
-H0)))))).
-
-theorem pc1_head_2:
- \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall
-(k: K).(pc1 (THead k u t1) (THead k u t2))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u:
-T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t))
-(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x:
-T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda
-(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead
-k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))).
-
-theorem pc1_t:
- \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2
-t3) \to (pc1 t1 t3)))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3:
-T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t:
-T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x:
-T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in
-(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1
-t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2
-x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1
-t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x
-x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1
-(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x
-H2))))) H4))))) H1)))))).
-
-theorem pc1_pr0_u2:
- \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0
-t2) \to (pc1 t1 t2)))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2:
-T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))).
-
-theorem pc1_head:
- \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall
-(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2
-t2))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead
-k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2)
-(pc1_head_2 t1 t2 H0 u2 k)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/arity_props.ma".
-
-include "LambdaDelta-1/nf2/fwd.ma".
-
-theorem pc3_dec:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c
-u1 u2) ((pc3 c u1 u2) \to False)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
-u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2
-t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T
-(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2)
-((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1
-x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4
-\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c
-u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda
-(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0)
-in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P:
-Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x
-x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let
-H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl
-(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10)))))
-(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1
-u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9
-in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t))
-False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2
-x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2
-in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t))
-False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1
-x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def
-(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def
-(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0
-H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))).
-
-theorem pc3_abst_dec:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2
-T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
-\to False))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
-u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2
-H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T
-(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T
-(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
-\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c
-x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T
-(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T
-(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
-\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2
-c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T
-(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x
-(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda
-(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u:
-T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_:
-T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c
-v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False)))
-(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0
-t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or
-(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2
-u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u)
-t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_:
-T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind
-Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead
-(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5
-(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t:
-T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def
-(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2
-T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
-\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead
-(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind
-Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda
-(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst)
-x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1
-x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda
-(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P:
-Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1
-(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead
-(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2)))
-(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1
-(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c
-u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t:
-T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False
-(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead
-(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t:
-T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda
-(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17
-\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def
-(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda
-(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20:
-(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c
-(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t
-x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4
-x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3)))
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead
-c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda
-(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3
-x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
-u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0:
-T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10
-(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t:
-T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def
-(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst)
-x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind
-Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in
-(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6)
-(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K
-(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False))))
-H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4)))))
-H12)))))) H9)))))) H6)))))) H3)))))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/defs.ma".
-
-definition pc3:
- C \to (T \to (T \to Prop))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3
-c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
-
-inductive pc3_left (c: C): T \to (T \to Prop) \def
-| pc3_left_r: \forall (t: T).(pc3_left c t t)
-| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
-| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/left.ma".
-
-include "LambdaDelta-1/fsubst0/defs.ma".
-
-include "LambdaDelta-1/csubst0/getl.ma".
-
-theorem pc3_pr2_fsubst0:
- \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
-t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
-c2 t2 t)))))))))))
-\def
- \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1
-t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i:
-nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2
-t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3
-t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0:
-(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0:
-T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda
-(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u))
-\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2
-t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr)
-u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2:
-T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c
-t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4
-w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0
-t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x:
-T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x
-t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3
-(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl
-u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e:
-C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3
-(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2
-t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e:
-C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2
-T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0
-t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3
-H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2:
-T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda
-(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0
-t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x
-H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c
-c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5))
-(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1))))))))))
-(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3:
-T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3
-t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
-T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr)
-u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2
-t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
-u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0
-H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0
-t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c
-c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
-u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def
-(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind
-(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8:
-(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i
-H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
-u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))
-(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11:
-(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
-(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
-(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u
-H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
-(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda
-(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3
-t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
-t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta
-c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
-u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3
-u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq
-C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
-i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
-x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
-u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let
-H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5)))
-H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
-i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0
-(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8))
-(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
-(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
-x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
-(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u
-H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
-i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda
-(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4
-t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
-t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta
-c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
-u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4
-u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6:
-(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c
-c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5:
-T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5:
-(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind
-Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8
-\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in
-(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b)
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))
-(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2
-c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5
-(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2
-t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1
-(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda
-(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow
-d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind
-x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
-(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u
-H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
-(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda
-(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3
-t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0
-t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5
-(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2
-(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0
-e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
-H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0
-(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b)
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
-i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
-x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
-u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let
-H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6)))
-H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
-i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
-(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0
-(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9))
-(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
-(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
-x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in
-(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def
-(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u
-H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
-i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda
-(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H22: (subst0 i x4
-t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0
-t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5
-(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2
-(pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0
-e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
-H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back t3 t0 u i H2 x4 u0
-(minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) (\lambda (H7:
-(le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0
-(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4)
-t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0
-H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4
-H3)))))))))))))))) c1 t1 t H)))).
-
-theorem pc3_pr2_fsubst0_back:
- \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
-t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
-c2 t t2)))))))))))
-\def
- \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t
-t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i:
-nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2
-t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0
-t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0:
-(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0:
-T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda
-(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u))
-\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3
-t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr)
-u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4
-(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0:
-C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c
-(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0))))))
-(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda
-(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e
-(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0
-t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e
-(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1))))))))))
-(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3:
-T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3
-t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
-T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr)
-u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0
-t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
-u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3
-H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3
-(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0
-(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c
-c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
-u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def
-(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind
-(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8:
-(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i
-H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
-u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))
-(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11:
-(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
-(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
-(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u
-H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
-(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda
-(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3
-t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
-t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta
-c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
-u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3
-u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq
-C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
-i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
-x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
-u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let
-H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5)))
-H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
-i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0
-(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8))
-(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
-(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0)
-x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
-(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u
-H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
-i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda
-(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4
-t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x
-t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta
-c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr)
-u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4
-u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6:
-(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c
-c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5:
-T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5:
-(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind
-Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8
-\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in
-(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b)
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))
-(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u
-c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3
-(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
-(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
-u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1
-(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda
-(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow
-d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind
-x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
-(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u
-H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3
-(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda
-(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3
-t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x
-t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta
-c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr)
-u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0
-i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6)
-t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0
-(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b)
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
-i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
-x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind
-Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr)
-u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let
-H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6)))
-H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus
-i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2
-(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0
-e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
-H6) t0 t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9:
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b)
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
-(pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
-(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
-x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13:
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in
-(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def
-(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u
-H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S
-i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda
-(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S
-i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4
-t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let
-H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n:
-nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x
-t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
-(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
-(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3
-t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8)))
-(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i
-(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0
-H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n
-i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5
-H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))).
-
-theorem pc3_fsubst0:
- \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
-t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
-c2 t2 t)))))))))))
-\def
- \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1
-t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i:
-nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2
-t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3
-t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2:
-C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u
-c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e
-(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0
-i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr)
-u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3
-H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e:
-C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0)))))
-(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda
-(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e
-(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i
-(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3
-H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0:
-(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2:
-((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4:
-T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind
-Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u:
-T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2
-t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e:
-C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5:
-T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1
-(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1
-t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda
-(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e
-(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0
-(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2
-c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda
-(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6:
-(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0
-t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2
-(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0:
-T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda
-(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u:
-T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall
-(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4
-t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4:
-T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda
-(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u))
-\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2
-t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr)
-u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1
-t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda
-(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e
-(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0
-t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0
-(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4:
-(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1
-c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr)
-u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0
-t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst
-i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/props.ma".
-
-include "LambdaDelta-1/pr3/fwd.ma".
-
-theorem pc3_gen_sort:
- \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort
-n)) \to (eq nat m n))))
-\def
- \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c
-(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c
-(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x:
-T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let
-H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n
-H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat
-(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0)
-\Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m]))
-(TSort m) (TSort n) H3) in H4))))) H0))))).
-
-theorem pc3_gen_abst:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall
-(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to
-(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
-t1 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
-t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst)
-u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c
-u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))))
-(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda
-(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2
-x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u:
-T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2
-x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3)))))
-(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
-t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead
-(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def
-(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead
-(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _)
-\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3)
-(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 |
-(TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst)
-x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let
-H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2
-(\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b:
-B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16
-u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1
-(H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))).
-
-theorem pc3_gen_abst_shift:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c
-(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind
-Abst) u) t1 t2)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def
-(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u)
-(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3
-(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2:
-((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2
-Abst u))) H0))))))).
-
-theorem pc3_gen_lift:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall
-(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d
-c e) \to (pc3 e t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e:
-C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t:
-T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e
-t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3:
-(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in
-(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e
-t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d
-x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e
-H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
-T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift
-h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t:
-T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1
-(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e
-t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
-
-theorem pc3_gen_not_abst:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1:
-T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b)
-u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S
-O) O (THead (Bind Abst) u2 t2))))))))))
-\def
- \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
-(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2:
-T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead
-c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda
-(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr)
-u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t:
-T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind
-Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind
-Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1)
-x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
-(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr)
-u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead
-c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5:
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
-(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1)
-t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1
-x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def
-(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
-c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
-T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1
-(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2
-x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
-u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind
-Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T
-(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
-_) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
-H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1
-(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
-t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2
-t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
-Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0:
-B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def
-(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O
-t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1)
-t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind
-Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind
-Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0
-x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6)))
-H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c:
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
-t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda
-(_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst)
-u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void
-Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead
-(Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c
-(THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2
-t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2
-t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1)
-x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
-(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall
-(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1)
-t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead
-(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
-c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
-T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c
-(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0
-x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u:
-T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x
-H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
-Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
-u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind
-Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x
-(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_:
-((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let
-H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6
-(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst)
-x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0)
-\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
-_) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3
-(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
-H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1
-(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
-t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2
-t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
-Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0:
-B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def
-(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O
-t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1)
-t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind
-Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind
-Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0
-x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6)))
-H4))))) H1))))))))) b).
-
-theorem pc3_gen_lift_abst:
- \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall
-(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2))
-\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda
-(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
-T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
-t1)))))))))))))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind
-Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H
-in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3
-c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1:
-T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
-T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1)))))))
-(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c
-(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0)
-in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3
-e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind
-Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1))))
-(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T
-x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2
-t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e
-t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2
-(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1:
-T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1
-x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x
-(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1:
-T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1:
-T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1:
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
-t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead
-(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14:
-(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0:
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10
-(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c
-u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0:
-T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T
-(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1))))
-(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_:
-T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1
-x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
-
-theorem pc3_gen_sort_abst:
- \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c
-(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda
-(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0
-\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0:
-T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c
-(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def
-(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
-T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u
-x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
-u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n)))
-(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind
-T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P
-H8)))))))) H3))))) H0))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/props.ma".
-
-theorem pc3_ind_left__pc3_left_pr3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to
-(pc3_left c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda
-(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2
-c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2:
-(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
-
-theorem pc3_ind_left__pc3_left_trans:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
-(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3:
-T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda
-(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
-t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3
-t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0
-t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0:
-(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda
-(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0
-t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0
-t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
-
-theorem pc3_ind_left__pc3_left_sym:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
-(pc3_left c t2 t1))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t)))
-(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
-(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
-t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3
-H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
-(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4
-t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3
-(pc3_left_r c t3))))))))) t1 t2 H)))).
-
-theorem pc3_ind_left__pc3_left_pc3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to
-(pc3_left c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
-t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
-T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1
-x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x
-(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x
-(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
-
-theorem pc3_ind_left__pc3_pc3_left:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
-(pc3 c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda
-(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c
-t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3
-c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
-(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3
-(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
-
-theorem pc3_ind_left:
- \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t
-t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
-T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1:
-T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to
-((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c
-t t0) \to (P t t0))))))))
-\def
- \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall
-(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1
-t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1
-t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
-\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2
-t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t
-t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda
-(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3:
-T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3
-(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left
-c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3
-(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0
-(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/defs.ma".
-
-include "LambdaDelta-1/nf2/pr3.ma".
-
-theorem pc3_nf2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
-t1) \to ((nf2 c t2) \to (eq T t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
-t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in
-(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T
-t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2
-x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x
-(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x
-(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c
-t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1
-H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2
-H_y0))))))))) H2))))))).
-
-theorem pc3_nf2_unfold:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
-t2) \to (pr3 c t1 t2)))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
-t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t:
-T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x:
-T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def
-(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t:
-T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/defs.ma".
-
-include "LambdaDelta-1/pc1/defs.ma".
-
-include "LambdaDelta-1/pr3/pr1.ma".
-
-theorem pc3_pc1:
- \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1
-t2))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c:
-C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t:
-T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1
-x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t))
-(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c)))))
-H0))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/defs.ma".
-
-include "LambdaDelta-1/pr3/pr3.ma".
-
-theorem clear_pc3_trans:
- \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to
-(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
-\def
- \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind
-T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1
-t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2
-x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2
-t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1
-H0))))) H1))))))).
-
-theorem pc3_pr2_r:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
-t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
-
-theorem pc3_pr2_x:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2
-t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
-t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
-
-theorem pc3_pr3_r:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
-t2 H (pr3_refl c t2))))).
-
-theorem pc3_pr3_x:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2
-t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
-t1 (pr3_refl c t1) H)))).
-
-theorem pc3_pr3_t:
- \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall
-(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1
-t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t:
-T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
-
-theorem pc3_refl:
- \forall (c: C).(\forall (t: T).(pc3 c t t))
-\def
- \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0))
-(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
-
-theorem pc3_s:
- \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c
-t2 t1))))
-\def
- \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1
-t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
-T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1
-x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t))
-(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
-
-theorem pc3_thin_dx:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
-(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u
-t2)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
-t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda
-(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u
-t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1
-x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead
-(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead
-(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f)))))
-H0))))))).
-
-theorem pc3_head_1:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
-(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
-u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda
-(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t)
-(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2:
-(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0))
-(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x
-H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl
-(CHead c k x) t)))))) H0))))))).
-
-theorem pc3_head_2:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
-(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u
-t2)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T
-(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u)
-t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1:
-(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2
-T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u
-t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1)
-(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
-
-theorem pc3_pr2_u:
- \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall
-(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
-\def
- \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
-(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
-t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
-x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t))
-x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
-
-theorem pc3_t:
- \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall
-(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
-\def
- \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
-(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
-t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
-x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
-T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1
-x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t))
-(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7:
-(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c
-H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2)))))
-H4))))) H1))))))).
-
-theorem pc3_pr2_u2:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
-(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
-t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x
-c t1 t0 H) t2 H0)))))).
-
-theorem pc3_pr3_conf:
- \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall
-(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t
-t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c
-t2 t H0) t1 H)))))).
-
-theorem pc3_head_12:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
-(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
-(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c
-u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
-
-theorem pc3_head_21:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
-(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
-(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c
-u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
-
-theorem pc3_pr0_pr2_t:
- \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c:
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
-(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
-t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
-(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3:
-(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0
-(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k
-u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
-(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k
-u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in
-(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr)
-u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k
-u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1)
-t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr)
-u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead
-c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda
-(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr)
-u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
-(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d
-(Bind Abbr) u) u2 H10)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match
-e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _
-k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in
-((let H13 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead
-d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind
-Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d
-c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13)
-in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t))
-(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0))
-(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O
-u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x
-(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c
-u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr)
-u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2
-t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10:
-(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans
-(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t
-(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9))
-(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f
-c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2)
-(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0
-(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3
-(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d
-(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0:
-K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4
-t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind
-Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_:
-(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
-t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind
-b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t
-(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d
-(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_:
-(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
-t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat
-f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t
-(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f
-u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7
-H4)))))))))))))) y t1 t2 H1))) H0)))))))).
-
-theorem pc3_pr2_pr2_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2
-u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
-T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3
-(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
-K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0
-t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
-(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
-(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1:
-C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y:
-C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
-T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4
-t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
-t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4
-t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
-C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
-(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
-t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
-c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
-(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n:
-nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
-t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1)
-(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
-(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3
-(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
-(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0
-| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind
-b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14
-\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
-with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1)
-(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
-Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr)
-u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let
-H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in
-(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind
-T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3
-(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2
-t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5
-t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind
-Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda
-(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus
-i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat
-(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in
-(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t)
-c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind
-Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind
-Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24))))))))
-(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O
-H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear
-(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0
-(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta
-(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11))
-(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0)
-(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O
-(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda
-(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0
-t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead
-c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5
-t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0))
-\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r
-(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t)
-t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0
-(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda
-(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead
-c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1)
-H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1
-i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1
-H)))).
-
-theorem pc3_pr2_pr3_t:
- \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
-(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to
-(pc3 (CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
-(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3
-(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
-u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
-(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1)
-\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2
-u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
-u1 H3)))))))))) t1 t2 H)))))).
-
-theorem pc3_pr3_pc3_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall
-(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2
-u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
-(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1
-t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
-K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
-(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
-t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3
-(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0:
-T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0
-t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c
-k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2)
-t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6:
-(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0
-x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2
-H0)))))) H4))))))))))))) u2 u1 H)))).
-
-theorem pc3_lift:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
-d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift
-h d t1) (lift h d t2)))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1
-t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t:
-T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda
-(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1)
-(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H
-t2 x H3))))) H1))))))))).
-
-theorem pc3_eta:
- \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t
-(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead
-(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H:
-(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v
-w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
-(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef
-O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl)
-(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead
-(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t)
-(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S
-O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u)
-H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w
-(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u))))
-(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
-(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl
-c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/props.ma".
-
-include "LambdaDelta-1/pr3/subst1.ma".
-
-theorem pc3_gen_cabbr:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
-(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
-\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
-a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall
-(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2))))))))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
-t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d
-c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c
-a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1:
-T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda
-(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda
-(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x:
-T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T
-(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2
-x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d
-x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x
-(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda
-(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3
-a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0
-(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2
-H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3)))))
-(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/props.ma".
-
-include "LambdaDelta-1/wcpr0/getl.ma".
-
-theorem pc3_wcpr0__pc3_wcpr0_t_aux:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall
-(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3
-(CHead c2 k u) t1 t2))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k:
-K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
-(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0:
-T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t))
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4
-t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3:
-(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead
-c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4
-t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c:
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead
-c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda
-(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2
-k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda
-(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d
-(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6
-t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c
-(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead
-d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2:
-C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
-T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda
-(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7:
-T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6
-t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t
-x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6
-t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x
-H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k
-u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr)
-H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
-
-theorem pc3_wcpr0_t:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
-T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1
-t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0:
-C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1:
-T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def
-(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T
-(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k
-u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3
-(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x
-(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2
-(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x
-H6)))))) H4))))))))))))) c1 c2 H))).
-
-theorem pc3_wcpr0:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
-T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind
-T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1
-t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2
-x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2
-(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/fwd.ma".
-
-include "LambdaDelta-1/subst0/dec.ma".
-
-include "LambdaDelta-1/T/dec.ma".
-
-include "LambdaDelta-1/T/props.ma".
-
-theorem nf0_dec:
- \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 t1 t2))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to
-(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl
-(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T
-(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda
-(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n)
-t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
-(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl
-(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T
-(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda
-(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n)
-t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T
-(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t:
-T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0
-t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or
-(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2)))
-(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b:
-B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0)
-t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T
-(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
-(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind
-Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2:
-T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda
-(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in
-(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O)
-O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead
-(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
-(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t
-t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0
-(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T
-(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S
-O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2)
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0)
-t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind
-Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let
-H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
-\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O
-x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S
-O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6
-P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3)))
-(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x)
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3)
-\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2)
-t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O)
-O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind
-Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t
-(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S
-O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2)))
-H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t
-t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t
-t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
-T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0
-t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0
-t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0
-(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda
-(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
-(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))
-(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda
-(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0
-t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def
-(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0
-H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind
-Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3:
-T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3:
-T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead
-(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3))
-(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0
-t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2:
-T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))
-(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead
-(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t
-t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst)
-t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P:
-Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0
-(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind
-Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t
-x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0)
-(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2:
-T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2:
-T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal
-T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3)))
-(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
-t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
-(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))))
-(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P:
-Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead
-(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind
-Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x
-t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
-\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0)
-(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2:
-T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T
-t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P))))))
-(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x
-\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or
-(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
-(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0
-(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift
-(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T
-(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind
-Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
-(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let
-H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to
-(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead
-(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
-(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2)
-\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2)
-\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead
-(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda
-(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))
-(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t
-t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0
-t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t
-x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def
-(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12
-t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead
-(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda
-(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3:
-T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead
-(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3))
-(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9:
-(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let
-H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3
-(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq
-T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x)
-(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
-(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead
-(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2
-H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T
-t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
-(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P:
-Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0
-(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind
-Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void)
-t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0)
-(THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2:
-T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2:
-T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal
-T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7))
-H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
-t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
-(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P:
-Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0
-(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind
-Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void)
-x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
-\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0)
-(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2:
-T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq
-T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t)
-P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4)))
-(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda
-(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda
-(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2
-t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2:
-T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead
-(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t
-t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void)
-t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S
-O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S
-O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t
-(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
-(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead
-(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y
-(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void
-not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f:
-F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0)
-t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T
-(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
-(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def
-H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
-T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
-T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P:
-Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq
-T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
-Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
-(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda
-(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T
-(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
-u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T
-(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
-Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
-(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0
-(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T
-(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3:
-T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind
-x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t
-t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq
-T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3:
-T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall
-(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or
-(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to
-(eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2:
-T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2))
-t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2)
-t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T
-(THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0
-(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat
-Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T
-(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1
-x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead
-(Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
-(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1
-(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat
-Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl)
-(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead
-(Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind
-Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7))))
-(pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2
-(pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst)
-x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2:
-T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda
-(t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2:
-T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead
-(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T
-(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1
-x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead
-(Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
-(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2)
-(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead
-(Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat
-Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1
-t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2:
-T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2)
-t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind
-Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2))
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void)
-x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat
-Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T
-(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1
-x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2))
-(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead
-(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P:
-Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow
-(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False
-| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
-\Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S
-O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst
-Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl
-x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w:
-T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P:
-Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq
-T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t
-t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
-T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0
-t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0
-t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0
-(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda
-(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
-(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0)
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))
-(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl)
-t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead
-(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0
-x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def
-(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def
-(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0
-H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0)
-in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl)
-t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3:
-T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t
-t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq
-T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0
-x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t
-x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda
-(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0
-(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead
-(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3:
-T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b)
-w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in
-(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat
-Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind
-Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t
-(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2
-H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
-(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b)
-y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat
-Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift
-(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda
-(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2:
-T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))
-(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not
-(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda
-(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
-x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0
-x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
-x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def
-(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3
-t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda
-(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead
-(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10)
-in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat
-Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))))
-(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1
-x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4
-(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11)))))))))))))
-H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2:
-T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0
-t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P)))
-(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t
-t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
-T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T
-t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror
-(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat
-Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2)
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0)
-t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))
-(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead
-(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat
-Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x
-(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x
-(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in
-(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat
-Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda
-(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))
-(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead
-(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t
-t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl)
-t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P:
-Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead
-(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
-(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T
-(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat
-Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x
-t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
-\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0)
-(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2:
-T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq
-T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t)
-P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3)))
-H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq
-T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
-Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
-(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat
-Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
-(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0)
-t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0
-(pr0_refl t0) t))) f)) k)))))) t1).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/defs.ma".
-
-inductive pr0: T \to (T \to Prop) \def
-| pr0_refl: \forall (t: T).(pr0 t t)
-| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1:
-T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1)
-(THead k u2 t2))))))))
-| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to
-(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1
-(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2))))))))
-| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1:
-T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0
-u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead
-(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t2)))))))))))))
-| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1:
-T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to
-(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
-| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall
-(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O
-t1)) t2))))))
-| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u:
-T).(pr0 (THead (Flat Cast) u t1) t2)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/props.ma".
-
-theorem pr0_gen_sort:
- \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
-\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq
-T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda
-(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0:
-T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T
-t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in
-(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n))
-t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
-t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let
-H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
-(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u:
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_:
-(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
-t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1))
-(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2
-t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda
-(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2
-v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
-t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
-(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1
-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind
-b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
-(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2
-u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda
-(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_:
-(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort
-n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
-H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1))
-H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1
-(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead
-(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind
-b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2
-(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda
-(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq
-T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1)
-(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x
-H0))) H))).
-
-theorem pr0_gen_lref:
- \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
-\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq
-T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda
-(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0:
-T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T
-t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n))
-t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
-t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let
-H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in
-(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u:
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_:
-(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
-t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1))
-(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2
-t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda
-(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2
-v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
-t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
-(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1
-t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind
-b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
-(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2
-u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda
-(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_:
-(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef
-n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
-H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1))
-H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1
-(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead
-(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind
-b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2
-(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda
-(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq
-T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1)
-(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x
-H0))) H))).
-
-theorem pr0_gen_abst:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1
-t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t:
-T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y:
-T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T
-t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda
-(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def
-(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r
-T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind
-Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1
-(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t
-H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda
-(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0)
-(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind
-Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind
-Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind
-Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind
-Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead
-(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T
-u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
-T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10))))
-H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1
-t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1
-t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl)
-v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind
-T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda
-(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
-v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
-u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind
-b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat
-Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda
-(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))
-H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
-u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2
-w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1
-t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind
-Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
-Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
-(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O
-t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in
-((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
-t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
-\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
-(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
-\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow
-(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda
-(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
-t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let
-H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t))
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T
-(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12
-\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_:
-False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1
-H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_:
-(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u
-t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
-t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))).
-
-theorem pr0_gen_appl:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1
-t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b)
-v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2))))))))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t:
-T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y:
-T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T
-t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
-t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t:
-T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T
-T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead
-(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
-t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
-T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl
-t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0
-u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind
-Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl)
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq
-T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
-(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
-(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
-(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0
-u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda
-(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2
-t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq
-T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
-t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
-T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let
-H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def
-(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
-T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind
-T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda
-(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl)
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
-T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda
-(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
-(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2
-(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7))
-H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda
-(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl)
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
-T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind
-b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1
-t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _)
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat
-Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u
-t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t)
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat
-Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1
-(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1
-(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1
-(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12
-\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
-b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in
-(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda
-(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl)
-(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead
-(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind
-Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2)
-t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0
-y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
-v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2
-(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2
-t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not
-(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1
-v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda
-(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl)
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1
-t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat
-Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _)
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat
-Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0)
-| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow
-t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1)
-H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t:
-T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1
-(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1
-(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def
-(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
-T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in
-(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1
-t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead
-(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in
-(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda
-(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat
-Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat
-Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat
-Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
-Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl)
-(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0))
-(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)))
-H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2:
-T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1
-t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1
-t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_:
-(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead
-(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2
-w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not
-(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead
-(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def
-(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in
-T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1
-t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq
-T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T
-(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
-(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
-True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
-
-theorem pr0_gen_cast:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t:
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))
-(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda
-(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat
-Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat
-Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0:
-T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat
-Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead
-(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
-u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T
-(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda
-(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0
-(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
-u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda
-(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0)
-(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat
-Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat
-Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat
-Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat
-Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0
-(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0
-(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda
-(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
-T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0
-t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2
-t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
-(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
-v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
-(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
-\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2
-t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
-Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda
-(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
-u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead
-(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
-\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
-True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in
-(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda
-(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast)
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead
-(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq
-T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
-t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T
-(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T
-(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b:
-B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1
-t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u:
-T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat
-Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0:
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0
-(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
-t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
-(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
-\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0)
-(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
-u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))).
-
-theorem pr0_gen_abbr:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t:
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda
-(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S
-O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t:
-T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
-T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t:
-T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T
-T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead
-(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0
-t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0
-t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
-t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind
-Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0:
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t
-H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda
-(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0
-t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0
-t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O
-t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind
-Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
-(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
-H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0
-| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
-H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
-| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
-H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind
-Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let
-H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr)
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4
-t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in
-(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2
-u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in
-(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind
-Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
-t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind
-Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7))
-H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_:
-(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O
-v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
-(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
-t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0:
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S
-O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_:
-(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1
-v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
-T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0:
-T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead
-(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
-T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0:
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S
-O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq
-T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1
-t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0:
-T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0
-(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0
-t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3:
-(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O
-t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq
-T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
-(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in
-(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T
-t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
-t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t:
-T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T
-t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
-t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t:
-T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or
-(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O
-u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro
-T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind
-Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1
-y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7)))))))))))))
-(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda
-(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind
-Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0:
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S
-O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S
-O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in
-((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
-t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
-\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
-(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
-\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow
-(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda
-(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
-t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let
-H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t))
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0))
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3
-(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0))
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))
-(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0:
-T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3)))))))
-(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1
-H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_:
-(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O
-t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
-(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))
-H4)))))))) y x H0))) H)))).
-
-theorem pr0_gen_void:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t:
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O)
-O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t:
-T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda
-(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda
-(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void)
-u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
-T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
-t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
-(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
-(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1
-t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2:
-T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
-u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda
-(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal
-T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
-(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
-(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
-(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0
-u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda
-(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2
-t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
-(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t
-(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
-(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t:
-T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T
-t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
-(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
-T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0
-t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void)
-u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
-(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
-(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead
-(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead
-(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B
-b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1
-v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda
-(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void)
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
-u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
-(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl)
-v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T
-(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda
-(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
-(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)))))
-H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
-u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda
-(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void)
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
-t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T
-(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T
-(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat
-_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w)
-(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead
-(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B
-b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
-t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda
-(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1
-t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
-(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k
-_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O
-t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
-(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u
-(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1
-t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let
-H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t))
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O)
-O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift
-(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2
-H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1
-t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
-t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
-(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x
-H0))) H)))).
-
-theorem pr0_gen_lift:
- \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0
-(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
-(t2: T).(pr0 t1 t2)))))))
-\def
- \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t
-x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
-(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat
-d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T
-x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t:
-T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq
-T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t:
-T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1
-x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2:
-T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq
-T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0))))))
-(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2:
-((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T
-(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0
-t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2
-t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
-x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
-T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda
-(k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4:
-T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))
-(\lambda (b: B).(\lambda (H6: (eq T (THead (Bind b) u1 t2) (lift h x1
-x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
-b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
-(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
-(t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0
-x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
-(Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T
-t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t:
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h
-(S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T
-(THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b)
-x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1)
-x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) x4) (\lambda (t:
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T (\lambda (t4:
-T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda
-(t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda
-(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T
-(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b)
-t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b)
-x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1
-x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind
-b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h x1 (THead (Bind b) x5
-x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift_bind b x5 x4 h
-x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 H_x0)))) (H2 x2 x1 H8)) t3
-H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind b u1 t2 x0 h x1 H6))))
-(\lambda (f: F).(\lambda (H6: (eq T (THead (Flat f) u1 t2) (lift h x1
-x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
-f) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
-(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda
-(t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0
-x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
-(Flat f) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T
-t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T
-(\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4:
-T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
-(\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f)
-u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4)))
-(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H10: (pr0
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-(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f)
-u2 (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2
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-(H11: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda
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-(t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq
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-(t4: T).(pr0 (THead (Flat f) x2 x3) t4)) (THead (Flat f) x5 x4) (sym_eq T
-(lift h x1 (THead (Flat f) x5 x4)) (THead (Flat f) (lift h x1 x5) (lift h x1
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-H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat
-f u1 t2 x0 h x1 H6)))) k H5))))))))))))) (\lambda (u: T).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0:
-T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
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-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall
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-(x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead
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-(z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_:
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-(Bind Abst) u t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind
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-h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0:
-T).(\lambda (z: T).(eq T x3 (THead (Bind Abst) y0 z)))) (\lambda (y0:
-T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind
-Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3)
-t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H9: (eq T x3 (THead (Bind
-Abst) x4 x5))).(\lambda (_: (eq T u (lift h x1 x4))).(\lambda (H11: (eq T t2
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-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda
-(t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T
-(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda
-(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda
-(x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H12: (pr0 x5
-x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda (t4:
-T).(eq T (THead (Bind Abbr) v2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0
-(THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex2_ind T (\lambda
-(t4: T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T
-(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 (lift h (S x1) x6)) (lift h x1
-t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5))
-t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T v2 (lift h x1 x7))).(\lambda
-(H13: (pr0 x2 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda
-(t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6)) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))))
-(ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h
-(S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
-(THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h
-x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S
-x1) x6)) (lift_bind Abbr x7 x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2
-H_x0)))) (H2 x2 x1 H7)) t3 H_x)))) (H4 x5 (S x1) H11)) x3 H9))))))
-(lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat Appl v1
-(THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b: B).(\lambda
-(H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
-v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h
-x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2:
-T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
-u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1
-x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2:
-T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2
-t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
-x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
-T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T
-(THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T
-(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
-(\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (THead (Bind b) u1 t2) (lift h x1 z)))) (ex2 T
-(\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl) x2
-x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead
-(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3)
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
-(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0
-z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4:
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
-x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4:
-T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda
-(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1)
-x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4:
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
-x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T
-(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4))
-(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
-(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift
-h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6)
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
-(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq
-T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4:
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S
-x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead
-(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1
-x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t:
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift
-(S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0
-(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4:
-T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda
-(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O
-v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
-Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T
-v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8)
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7)
-(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4)))
-(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))))
-(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T
-(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t
-(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
-Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat
-Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
-(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0
-(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda
-(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat
-Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
-(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat
-Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead
-(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S
-x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat
-Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5
-x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1)
-x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1
-x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2
-H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12))))))
-(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1
-(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1:
-T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0:
-T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2:
-T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w:
-T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1
-x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
-Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
-(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
-(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
-x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
-(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9:
-(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda
-(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1
-t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3
-(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4:
-T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
-(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3
-(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3
-(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T
-(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2
-T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda
-(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0:
-(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1
-x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w)
-(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let
-H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11
-(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in
-(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5)
-(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq
-T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2
-T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1
-t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6:
-T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17:
-(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t:
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h
-x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T
-(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O
-x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3)
-t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5
-x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6))
-(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17))
-w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1
-H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1
-H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u:
-T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u
-(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda
-(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
-T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2)
-(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
-(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
-(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1
-x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T
-(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift
-h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S
-O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1))
-(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1)
-(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O))
-H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4:
-T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1
-t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4:
-T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift
-h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4:
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t)
-t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
-T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda
-(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5:
-T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4
-x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
-t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O
-x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1
-t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5
-(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4
-x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0
-H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4:
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u:
-T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast)
-u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T
-x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift
-h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast)
-x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h
-x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
-(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0
-x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
-T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T
-t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4)
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda
-(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4:
-T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
-Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3
-H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1
-H3)))))))))) y x H0))))) H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/fwd.ma".
-
-include "LambdaDelta-1/lift/tlt.ma".
-
-theorem pr0_confluence__pr0_cong_upsilon_refl:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3:
-T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to
-(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x)
-\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4))
-t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
-v2) t5)) t)))))))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda
-(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda
-(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda
-(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0
-(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead
-(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3
-(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4
-t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5)
-(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S
-O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind
-b))))))))))))))).
-
-theorem pr0_confluence__pr0_cong_upsilon_cong:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2:
-T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall
-(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5:
-T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t))
-(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
-t5)) t)))))))))))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda
-(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2
-x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2
-x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda
-(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t))
-(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
-t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0))
-(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat
-Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp
-(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat
-Appl)) (Bind b))))))))))))))))))).
-
-theorem pr0_confluence__pr0_cong_upsilon_delta:
- (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w:
-T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x:
-T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2
-x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to
-((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead
-(Flat Appl) (lift (S O) O v2) t5)) t))))))))))))))))))))
-\def
- \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2:
-T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2:
-T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2:
-(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2
-x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda
-(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T
-(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t))
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O
-v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0
-(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0
-(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead
-(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H
-u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O
-v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2)
-(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind
-Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2:
-T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda
-(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl)
-u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3
-(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8:
-(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t:
-T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t:
-T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))
-(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon
-Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift
-(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O)
-O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl))
-(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9
-(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1
-H5))))))))))))))))))).
-
-theorem pr0_confluence__pr0_cong_upsilon_zeta:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3:
-T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0
-u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1:
-T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat
-Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
-(lift (S O) O v2) (lift (S O) O x))) t)))))))))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda
-(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda
-(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x:
-T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4:
-(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t:
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0:
-T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead
-(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O
-(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1
-t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat
-Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl)
-(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O)
-O)))))))))))))))).
-
-theorem pr0_confluence__pr0_cong_delta:
- \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to
-(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall
-(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda
-(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind
-Abbr) u3 w) t))))))))))))))
-\def
- \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3
-t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda
-(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3
-x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2:
-T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t:
-T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
-u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead
-(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))
-(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3
-x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2))
-(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w
-w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead
-(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)))
-(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0
-x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda
-(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta
-u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4))
-(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).
-
-theorem pr0_confluence__pr0_upsilon_upsilon:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2:
-T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1:
-T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to
-(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2
-x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl)
-(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t2)) t)))))))))))))))))))
-\def
- \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
-(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2
-x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1
-x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T
-(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1)
-t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0)
-x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat
-Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0)
-(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1
-H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O
-x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S
-O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).
-
-theorem pr0_confluence__pr0_delta_delta:
- \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to
-(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to
-(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0)
-\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))))))))))))))))
-\def
- \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2
-t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0:
-(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2:
-(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5
-x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2:
-T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
-t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0
-x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2:
-T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
-t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w
-x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
-(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp
-u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr))))
-(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O
-x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0
-O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
-(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7:
-(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t:
-T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
-u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr))
-(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x
-H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2:
-T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda
-(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2
-w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1:
-T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0
-w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0
-w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
-T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
-(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1
-H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda
-(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T
-(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
-(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda
-(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t:
-T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1)
-(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2
-x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in
-(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
-T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x
-H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda
-(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x
-x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t:
-T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
-t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3:
-T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1
-x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
-(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta
-u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11))
-(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead
-(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))
-(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2
-w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
-(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9
-(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0
-x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5))
-(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
-
-theorem pr0_confluence__pr0_delta_tau:
- \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to
-(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2
-t)))))))))
-\def
- \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2
-t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda
-(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda
-(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
-(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S
-O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t:
-T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w
-(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
-(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda
-(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))))))))
-(pr0_gen_lift t4 t3 (S O) O H0)))))))).
-
-theorem pr0_confluence:
- \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0
-t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))
-\def
- \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to
-(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3))
-(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall
-(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0
-v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2
-t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2:
-T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda
-(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4
-t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2
-t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3
-t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1)
-\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))))
-(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5:
-T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0
-return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4
-t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6:
-T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4
-t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2)
-\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))
-(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6:
-T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t
-(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda
-(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5:
-T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5:
-T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall
-(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8:
-T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5:
-T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall
-(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8:
-T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T
-(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def
-(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T
-(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1)
-(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5)
-H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead
-k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1
-t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4
-t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
-t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5)
-(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1
-u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6:
-T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda
-(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k
-u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2
-(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall
-(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8:
-T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0
-t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0
-(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2
-t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12))))
-t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow
-(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4))
-t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr)
-v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0
-t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind
-Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0
-v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
-(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4
-t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead
-(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl)
-v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def
-(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead
-(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6:
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
-T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7)
-in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind
-Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead
-(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind
-Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1
-v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl)
-v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1
-(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
-v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1
-t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B
-b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda
-(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1
-u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6:
-T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2
-T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t
-(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1
-t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v:
-T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v
-t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8
-t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in
-(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13
-(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta
-u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1
-t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead
-(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to
-((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T
-(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda
-(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda
-(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5
-w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead
-(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6:
-T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6))
-H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6:
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
-T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T
-(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0
-(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11
-t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10))
-t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7:
-(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5
-t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5
-t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5
-t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to
-(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))
-(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def
-(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O)
-O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6:
-T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let
-H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u
-(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6:
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
-T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in
-(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6))
-(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl
-t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5
-H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda
-(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T
-t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
-(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda
-(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t
-(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T
-(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6
-t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda
-(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def
-(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7:
-T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
-T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u
-t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6))
-(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1
-H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t)
-(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) |
-(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1
-t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3)
-(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4)
-\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))
-(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda
-(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5
-t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda
-(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5:
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0
-t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5
-t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0
-t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T
-(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))))
-(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1
-t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7:
-T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def
-(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let
-H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to
-(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T
-(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead
-k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6))
-(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k
-u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5
-(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow
-(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3
-t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6)
-t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead
-k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda
-(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5
-t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7))
-H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0
-t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1
-| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in
-((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _
-t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20:
-(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r
-K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7))
-(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1
-(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3
-(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0
-t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0
-u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x:
-T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda
-(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7:
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))
-(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3
-x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7:
-T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25
-k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2
-H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21)))))
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10)
-\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u
-t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead
-(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind
-Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
-(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2
-t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14:
-(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda
-(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind
-Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat
-Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
-(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let
-H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
-\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat
-Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in
-(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead
-k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let
-H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24
-\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5)
-H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8:
-T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T
-t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl
-t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda
-(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq
-T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T
-(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda
-(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda
-(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8:
-T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0
-(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0
-(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2
-x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead
-(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6
-H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4
-H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3
-H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead
-(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9)
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _)
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
-\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K
-(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead
-k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind
-Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda
-(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9
-u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2
-t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))))
-(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead
-(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda
-(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead
-(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8)
-t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to
-((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9)
-t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_:
-(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8
-t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat
-Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind
-Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38:
-(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2
-t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3
-t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0:
-T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x)
-(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind
-Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2
-H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat
-Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5)
-(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4
-H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0
-(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8
-H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind
-Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind
-Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind
-Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
-\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5)
-H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26)))
-| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda
-(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst)
-u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S
-O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead
-(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _
-_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5)
-H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
-v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26
-H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28:
-(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq
-T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr)
-u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
-\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T
-(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O
-u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26
-H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T
-(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda
-(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
-((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10)
-\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in
-lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match
-t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
-t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind
-Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S
-O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match
-k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u
-t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S
-O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0
-u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8
-t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10:
-T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind
-Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind
-T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst))
-\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4)
-t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda
-(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to
-((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4)
-t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda
-(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match
-(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
-(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead
-(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5
-H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25
-H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat
-Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let
-H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in
-(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21)))))
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0
-u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1
-(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1
-(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
-v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7:
-T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6)
-\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0
-t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1
-v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat
-Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda
-(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
-| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
-(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let
-H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
-\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0
-t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat
-Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in
-(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead
-k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
-(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7:
-T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7:
-T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in
-pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T
-t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7)
-\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30:
-(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4)
-\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))
-(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0
-t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8)
-t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8:
-T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat
-Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2
-x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16
-u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1
-(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind
-b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow
-(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32:
-(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7)
-(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 |
-(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7)
-(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
-(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7)
-(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4
-u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
-t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9:
-T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4)
-t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S
-O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda
-(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8)
-\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
-(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead
-(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T
-(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10))))))
-(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda
-(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9:
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))
-(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6
-x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9))
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8))
-t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44:
-(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0
-v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5
-t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
-(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda
-(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45
-H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1
-(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b)
-u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b)
-u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3
-H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1
-(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl)
-v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5
-H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33))
-H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31:
-(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0
-t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def
-(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T
-(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33))
-H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32)
-\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4
-t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead
-(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat
-Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0)
-u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst))
-\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b)
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31
-H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq
-T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T
-(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind
-Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9]))
-(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _)
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7)
-(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0)
-\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5)
-\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T
-u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind
-Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to
-(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10:
-T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq
-T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0
-O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10))
-(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O
-v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w)
-t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to
-((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead
-(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0
-u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43
-\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat
-Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to
-(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11))
-(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def
-(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr
-H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst)))
-H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9:
-T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0
-t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9))
-(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl)
-u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda
-(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9:
-T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0
-(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))
-(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2
-x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51
-t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead
-(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr)
-u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind
-Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40
-u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat
-Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5)
-(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19))))))))
-t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35))
-H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda
-(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0
-t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let
-rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9
-with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0
-u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))])
-in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match
-t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
-t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b)
-u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
-\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S
-O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda
-(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0
-| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in
-(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5)
-\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))
-(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O
-t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2
-T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10:
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O)
-O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8)
-\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
-(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not
-(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead
-(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b
-Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda
-(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0
-t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11)
-\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11
-t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5
-(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37)
-in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O)
-O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9)))
-(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl)
-u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
-(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O
-x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9:
-T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
-(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9))
-t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9))
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
-x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9))
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
-x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0
-v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48
-H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b)
-u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b)
-u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O)
-O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1
-(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44))))
-(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u
-(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) |
-(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u
-t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def
-(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0
-t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
-t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5))
-(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10
-H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12:
-(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr)
-u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead
-(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6
-w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w)
-t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to
-((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0
-u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
-(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind
-Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
-(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead
-(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K
-k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr)
-(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda
-(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1
-(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3
-(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0
-t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead
-(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))
-(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6
-x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7))
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0
-u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w
-H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2
-H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k
-H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5
-t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O
-t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O)
-O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5
-t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7:
-T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0
-(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not
-(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t
-(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O)
-O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind
-b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3)
-(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7]))
-(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20:
-(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O
-t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7:
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def
-(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind
-T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T
-(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7))
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7:
-T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O
-x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7:
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8:
-T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7:
-T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O
-x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0
-x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead
-(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta
-b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26
-t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18))
-H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u)
-\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11:
-(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
-t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8))
-(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2
-(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2
-t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let
-H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
-(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat
-Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
-(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead
-(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k
-(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast)
-(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda
-(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7
-u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8
-t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2
-t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda
-(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda
-(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2
-t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20
-t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15))))
-t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t)
-(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4
-H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst)
-u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead
-(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda
-(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T
-(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda
-(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5
-t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda
-(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5:
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
-T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5
-t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
-T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_:
-T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
-T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2))
-H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead
-(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda
-(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7))))
-(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat
-Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t
-(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7)
-\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9))
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind
-Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2
-t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u
-t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4))
-(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq
-T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda
-(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6)
-t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T
-(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2)
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4)
-t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda
-(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
-(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let
-H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) |
-(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
-(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat
-Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow
-(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat
-Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq
-T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda
-(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda
-(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda
-(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def
-(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3)
-H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8:
-T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T
-t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl
-t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda
-(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq
-T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T
-(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda
-(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda
-(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5
-(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u
-t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v:
-T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v
-t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10
-t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let
-H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T
-(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead
-(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30:
-(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0
-(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2
-(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4
-t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1
-(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6
-H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3
-H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead
-(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9)
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _)
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
-\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K
-(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead
-k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
-T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat
-Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda
-(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9
-u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2
-t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))))
-(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead
-(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda
-(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead
-(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8)
-t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to
-((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_:
-(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda
-(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19)
-in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9)
-\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to
-(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11
-t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let
-H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T
-(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda
-(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda
-(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0
-t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead
-(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8
-x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead
-(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0
-H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans
-(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3))
-(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst)
-u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind
-Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0
-(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24
-H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T
-(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u
-t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def
-(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T
-(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24
-v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat
-Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29:
-(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let
-H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
-K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not
-(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) |
-(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead
-(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead
-(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow
-(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match
-b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
-\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
-False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind
-Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) |
-(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind
-b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8
-t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
-\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10)
-\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in
-lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match
-t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
-t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind
-Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S
-O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match
-k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u
-t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S
-O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0
-u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8
-t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10:
-T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat
-Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind
-T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst))
-\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda
-(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to
-((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda
-(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match
-(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
-(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3
-H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24
-H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat
-Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let
-H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in
-(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21))))
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6
-H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind
-Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind
-T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T
-(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2
-t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T
-(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5
-t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0
-t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
-| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7)
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])]))
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead
-(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst)
-u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_:
-(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
-Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7
-v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4))
-H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0
-t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda
-(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25:
-(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2
-t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))
-(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3
-x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x)
-(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind
-Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2
-H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat
-Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5)
-(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15))))))))
-H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1
-u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0
-(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0
-(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0
-v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15:
-(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6))
-t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6))
-(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8))
-(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b
-Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
-t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl)
-v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1
-t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
-| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7)
-\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
-(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7)
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])]))
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead
-(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead
-_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3))
-(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u
-u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst
-(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
-(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O
-v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
-Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in
-False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat
-Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22))
-H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6
-H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5)
-t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
-Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
-(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda
-(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w)
-(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5
-t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda
-(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead
-(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)))
-H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0)
-\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5))
-t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O
-t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6)
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7:
-T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0
-(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_:
-(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t
-(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7))
-H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in
-T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2
-t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6
-H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5)
-t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
-(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq
-T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))
-(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0
-t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst)
-u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
-F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
-(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead
-(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T
-t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2)))))
-t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5)
-\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to
-((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
-t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2)
-\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6))
-(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b
-Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda
-(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5:
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5)
-\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind
-T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7:
-T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_:
-T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r
-T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b)
-u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17
-\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6:
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
-T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6)
-in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)))
-(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10
-(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13)
-H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T
-(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T
-(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3)
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
-(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda
-(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
-T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5
-t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) |
-(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1
-t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
-(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat
-Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead
-(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1
-u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0:
-K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26
-\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat
-Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19
-(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return
-(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead
-(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow
-(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7
-t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2
-T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))
-(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1
-t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat
-Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T
-(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32
-\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall
-(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda
-(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead
-(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1
-(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0
-v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
-T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x:
-T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0
-(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b
-H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl)
-u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead
-(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow
-(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31:
-(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7)
-(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 |
-(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7)
-(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
-(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7)
-(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4
-u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3
-t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9:
-T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to
-((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat
-Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda
-(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8)
-\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
-t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead
-(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T
-(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10))))))
-(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def
-(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26
-(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9:
-T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to
-(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12))
-(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind
-b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2))
-H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0
-u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3
-(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2
-x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9))
-(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45:
-(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5
-t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0
-(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda
-(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat
-Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x
-H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead
-(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx
-(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38
-u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl)
-u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat
-Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx
-(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7
-(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b)
-H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29)
-\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u
-t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8)
-t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7))
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow
-(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False
-| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind
-((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))
-H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31)
-\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4
-t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead
-(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat
-Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0)
-u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst))
-\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9))
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29
-H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31:
-(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T
-(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
-(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind
-Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9]))
-(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _)
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7)
-(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1)
-\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5)
-\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0
-(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda
-(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4
-u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind
-Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to
-(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
-t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq
-T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0
-O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat
-Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w)
-t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to
-((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead
-(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10:
-T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1
-u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42
-\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23
-Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
-Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T
-(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr
-H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat
-Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def
-(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10:
-T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12:
-T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat
-Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1
-(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0
-v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead
-(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x:
-T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda
-(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9:
-T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9))
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9)))
-(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9))
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead
-(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5
-x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead
-(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O)
-O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x
-H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind
-Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx
-(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5
-H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat
-Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3)
-(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12)))))
-(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3
-H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b
-H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u)
-\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead
-(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T
-\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T
-\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0
-d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
-t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b)
-u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
-\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S
-O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda
-(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0
-| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in
-(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3)
-\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T
-(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))
-(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O
-t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2
-T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10))))))))
-(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7)
-(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to
-(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
-t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not
-(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10:
-T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b
-Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda
-(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let
-H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t))
-H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t
-(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v
-t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10
-t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead
-(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda
-(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9:
-T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda
-(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x:
-T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7
-x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10))
-(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def
-(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda
-(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
-x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0:
-T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T
-(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda
-(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S
-O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda
-(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T
-(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta
-b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans
-(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind
-b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx
-(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42
-u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46
-u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8
-t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33))
-H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T
-(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8
-t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8
-t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead
-(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t
-H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda
-(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda
-(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0
-(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6)
-t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6)
-t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda
-(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t
-(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _)
-\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat
-Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match
-t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind
-Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead
-_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow
-t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0
-(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26:
-(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
-Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2))
-H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0
-(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda
-(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b
-(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match
-(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
-(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31))
-b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) |
-(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17:
-(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T
-(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T
-(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead
-(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0
-Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind
-b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind
-b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B
-b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0
-Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda
-(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
-(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead
-(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
-\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24)
-in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
-(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _
-_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
-in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
-\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24)
-in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead
-_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow
-t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0
-(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30:
-(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
-b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2))
-H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0
-(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda
-(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6))
-t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0
-H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29)
-in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in
-(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T
-(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl)
-(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4
-x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7))
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0
-(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda
-(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T
-(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda
-(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O)
-O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42:
-(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2
-u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead
-(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0
-t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0)
-u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3
-H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0
-(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat
-Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27))
-H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13
-t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5)
-t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind
-Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0
-u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda
-(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w)
-t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to
-((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
-T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5
-t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda
-(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead
-(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0
-(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) |
-(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind
-b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead
-(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B
-b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2
-t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not
-(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
-T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5
-t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in
-(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20)
-in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6
-(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u)
-\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15:
-(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
-t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2
-t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5
-t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_:
-(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
-Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in
-(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5)
-H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))
-H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t)
-(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2
-t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3)
-t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind
-Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1
-u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0
-t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind
-Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1
-u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0
-t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1
-u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11
-\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_:
-(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
-[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T
-t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))
-(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let
-H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr)
-u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6
-t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead
-(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6:
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
-T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T
-(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0
-(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead
-(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq
-T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12
-k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T
-(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead
-k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
-(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda
-(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0
-u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
-T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) |
-(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
-H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1
-| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
-H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
-H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr)
-k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))))
-(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13
-(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1
-(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3
-(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0
-t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead
-(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7)))
-(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6
-x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7))
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0
-u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3
-t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w
-H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5)
-u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6
-H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta
-u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1
-(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6)
-t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_:
-T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6)
-t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
-(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0
-t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr)
-u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let
-H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in
-(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13
-H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14)
-\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0
-t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S
-O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5))
-(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
-v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
-(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3
-(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7
-t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1
-v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead
-(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead
-(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b)
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15
-H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow
-(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T
-(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda
-(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6)
-\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr)
-u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead
-(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7:
-T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))
-(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19:
-(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T
-(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _)
-\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in
-((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _
-t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5)
-H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda
-(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to
-(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in
-(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let
-H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda
-(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28:
-(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3
-t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0
-u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w
-H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0
-t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6
-H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5
-t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O
-t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O)
-O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5
-t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda
-(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda
-(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))
-(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead
-(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr |
-(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18)
-in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead
-_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift
-(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3)
-(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1
-u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0:
-B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b
-(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr
-H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def
-(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind
-T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T
-(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7))
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
-T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O
-x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7:
-T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7:
-T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0:
-T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2
-x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl
-(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30
-t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6
-(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u)
-\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13:
-(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
-t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2
-w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))
-(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T
-(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17
-\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17))))
-t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t)
-(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3
-u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3))
-t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O
-t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4)
-\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))
-(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b
-Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda
-(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8:
-(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda
-(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
-[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5
-t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6:
-T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T
-(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t
-(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in
-(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to
-(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T
-(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead
-(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1
-t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1
-(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11)))
-t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow
-(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6)
-t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6)
-t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6)
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))
-(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r
-T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4
-(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
-_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
-(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7]))
-(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7:
-T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
-\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
-\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d
-u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
-O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
-(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort
-n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
-false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map
-f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
-O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O
-t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K
-(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
-t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r
-K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23
-\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19)
-in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7:
-T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
-(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S
-O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda
-(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let
-H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x)
-(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
-(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7))
-(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
-(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0:
-T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift
-(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx
-(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O)
-O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta
-u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1
-(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6)
-t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_:
-T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
-(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind
-Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_:
-(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda
-(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl)
-v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b)
-u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))
-H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2
-H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead
-(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat
-Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead
-(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat
-Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2)
-\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2
-(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not
-(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
-(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0)
-u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1
-t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7)))
-H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6
-H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5)
-t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
-Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
-t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind
-Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
-t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda
-(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r
-T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4
-(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in
-((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7
-_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr)
-u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
-((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in
-lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
-t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7)
-\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1
-t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24
-\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21)
-in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7:
-T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
-(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift
-(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda
-(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in
-(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6
-(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in
-(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w)
-t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1
-x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1)
-(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S
-O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1
-H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t
-H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda
-(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T
-t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq
-T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))
-(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3))
-t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal
-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
-in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
-\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0
-(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
-\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S
-O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7:
-T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
-\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
-\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1)
-(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O
-t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat)
-(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) |
-(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
-false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f
-d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
-O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O
-t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u
-u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7:
-T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall
-(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
-T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in
-(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3
-t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1
-Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
-t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))
-x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2
-H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) |
-(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0
-t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5)
-(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5
-t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u
-(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def
-(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in
-T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6
-t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4
-(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow
-(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4
-t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to
-((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
-t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0
-t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2
-t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return
-(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to
-((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
-t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5
-t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))
-(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t
-(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T
-(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1
-t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda
-(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def
-(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7:
-T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
-T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u
-t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
-(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2
-H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7
-t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda
-(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_:
-T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
-(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7:
-T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13:
-(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
-Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _)
-\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _)
-\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
-\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda
-(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast)
-(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
-(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T
-(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in
-(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in
-(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6)
-t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
-(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx
-(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9
-H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq
-T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T
-(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind
-Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0
-v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
-(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6)
-t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
-t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead
-(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T
-(THead (Flat Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
-(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
-True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H14) in
-(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
-(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b
-H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat
-Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl)
-v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
-v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
-t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B
-b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda
-(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1
-u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
-T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind
-b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1
-(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1
-t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta
-u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr)
-u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T
-(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2)
-\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T
-(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda
-(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_:
-(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let
-H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7))
-H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat
-Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
-t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8
-H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead
-(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T
-(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not
-(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2
-(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq
-B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda
-(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O)
-O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in
-(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
-t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6
-H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda
-(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T
-t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
-(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda
-(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
-T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t
-(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0
-t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast)
-u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3)
-(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8:
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10:
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast)
-u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5
-H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
-t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))
-(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))
-x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11))))))
-H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t)
-(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2
-(refl_equal T t) (refl_equal T t1))))))))) t0).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/defs.ma".
-
-include "LambdaDelta-1/subst0/subst0.ma".
-
-theorem pr0_lift:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall
-(d: nat).(pr0 (lift h d t1) (lift h d t2))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t)
-(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d:
-nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
-(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0
-(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0
-(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda
-(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t:
-T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2)
-(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k
-d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d)
-t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4))
-(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h
-d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_:
-(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h
-d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
-t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3)
-(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead
-(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u
-t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r
-T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s
-(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t)
-(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h
-d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl)
-(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s
-(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d)
-u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl)
-d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d
-(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s
-(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s
-(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3)))
-(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda
-(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d:
-nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d:
-nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d:
-nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d:
-nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d)
-(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b)
-(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3))
-(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead
-(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O)
-O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead
-(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d))
-t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O
-v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead
-(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift
-h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t)))
-(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h
-d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d
-u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4)))))
-(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat
-Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d)
-t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O)
-d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d
-u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d)
-t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d
-v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b)
-d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S
-O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3))
-(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
-v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3)
-h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
-u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1)
-(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
-t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3)
-(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda
-(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift
-h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr)
-u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr)
-d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind
-Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S
-d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in
-(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2)
-(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d
-(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal
-nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w))
-(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3))
-(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0:
-(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
-t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3)
-(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d:
-nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S
-O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d)
-(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O
-t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t:
-T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift
-h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S
-O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d)))
-(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift
-(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_:
-(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h
-d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d:
-nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d)
-t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast)
-d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u
-t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))).
-
-theorem pr0_subst0_back:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t:
-T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T
-(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3)))))))))
-(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1
-v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t:
-T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0)
-(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1:
-((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t))
-(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
-(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0
-u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x:
-T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T
-(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0
-(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3
-H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v:
-T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0
-(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T
-(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t
-t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind
-T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t
-(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4
-x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
-(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x)
-(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4:
-T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t:
-T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4:
-T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda
-(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T
-(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T
-(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t
-(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3
-x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t))
-(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1
-t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda
-(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda
-(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3
-t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3
-H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
-
-theorem pr0_subst0_fwd:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t:
-T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T
-(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4)))))))))
-(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v
-u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t:
-T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0)
-(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1:
-((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t))
-(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
-(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0
-u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x:
-T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T
-(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0
-(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3
-x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda
-(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_:
-(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to
-(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3
-t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind
-T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0
-(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4
-x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
-(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x)
-(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4:
-T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t:
-T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4:
-T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda
-(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T
-(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T
-(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead
-k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3
-x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t))
-(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1
-t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda
-(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda
-(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4)
-t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8
-t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
-
-theorem pr0_subst0:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall
-(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1
-v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v2 t2 w2))))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i:
-nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
-t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0
-w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i:
-nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1
-v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd
-v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0:
-(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
-nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
-u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2
-w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
-t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
-nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
-t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i:
-nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2:
-T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1
-(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5:
-T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))
-(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5))))
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4))
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k
-u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3
-t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq
-T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1
-(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1
-(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3)
-(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t
-w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2)
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))
-(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead
-k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda
-(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T
-(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2
-(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind
-T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0
-(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3)
-w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0:
-T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror
-(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x
-t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T
-(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2
-(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k)
-(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7))))
-H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5)))
-(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq
-T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0
-w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1
-(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k
-u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2:
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind
-(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k
-i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2:
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-(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T
-x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst)
-x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1))
-(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4)
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
-(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead
-(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2))
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-(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1))
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1
-(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
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-(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s
-(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2))
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-(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0
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-(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2)
-(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16
-v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1
-H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i)
-H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
-w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0
-v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead
-(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5:
-T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat
-Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2
-t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3
-(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1
-x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3)
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-(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T
-(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2:
-T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1
-(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u
-u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda
-(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_:
-(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t:
-T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in
-(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t:
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-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0
-x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
-w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
-Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4)
-w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0
-(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4
-H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2:
-T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda
-(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind
-Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead
-(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3
-(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0
-x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl)
-x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4)
-(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind
-Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10:
-(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5:
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda
-(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind
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-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead
-(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead
-(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i))
-v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat
-Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat
-Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i
-v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2:
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3
-t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
-Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4)
-w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2:
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat
-Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda
-(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
-t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0
-(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4))
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x))
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-(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2
-x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
-t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
-t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14)
-(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5)))
-(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0
-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2:
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3
-t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
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-w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s
-(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead
-(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T
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-(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x))
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0
-(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
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-(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
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-(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T
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-(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4))
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x))
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-(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2
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-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
-t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
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-(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8
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-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T
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-(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T
-x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst)
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-(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2:
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-(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5)))
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-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2
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-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0
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-(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x
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-x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1))
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-i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T
-x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat
-Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b)
-u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))
-(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)))
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x:
-T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s
-(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t:
-T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in
-(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or
-(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T
-(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat
-Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x))
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2:
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat
-Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))
-(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2
-x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
-b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift
-(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind
-b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl))
-u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2:
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s
-(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead
-(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)))
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x))
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x
-x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind
-(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i
-v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0
-(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b)
-u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead
-(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2
-H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O)
-O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl)
-v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))
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-x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
-b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift
-(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s
-(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4
-x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b)
-(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T
-(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda
-(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead
-(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i)
-v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat
-Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind
-b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda
-(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind
-T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b)
-x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3))
-(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O)
-O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind
-(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s
-(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead
-(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2
-x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2)
-(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl)
-i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2:
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat
-Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead
-(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2
-T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i
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-(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2
-x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
-b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift
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-T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s
-(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2
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-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2))
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda
-(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead
-(Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0
-(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0
-v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
-b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift
-(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H22 x2 x H20 x3 t4 H18) (subst0_fst
-v3 x u2 i H21 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda
-(H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind
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-(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i
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-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
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-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind
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-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift
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-v2) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H24 O (le_O_n i) b) t4 (Flat
-Appl)))))))) H22)) (H2 v0 x0 i H11 v3 H8))))) H19)) (H4 v0 x2 (s (Flat Appl)
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-(w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T
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-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2:
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-w2)))) (\lambda (x: T).(\lambda (H19: (pr0 x3 x)).(\lambda (H20: (subst0 (s
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-w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2:
-T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda
-(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2))
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x:
-T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T
-(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2
-w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0
-O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def
-(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in
-(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w
-x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2))
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x
-H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18))))))))
-(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2
-H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2:
-T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1
-w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind
-Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13:
-(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2:
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind
-Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead
-(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2
-w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2
-x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind
-Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x
-x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0
-(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead
-(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr)
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))
-(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd
-(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind
-Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14:
-(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2
-u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0
-x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O
-x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind
-Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4
-x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal
-nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20
-\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S
-i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t))
-(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1)
-w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda
-(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22:
-(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2))
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2
-H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4
-(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21)))))))
-(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S
-i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i
-H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7))
-(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1:
-T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2)
-\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda
-(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift
-(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T
-(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda
-(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b)
-u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2
-t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or
-(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i
-v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b)
-u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda
-(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6:
-(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u
-x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda
-(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5:
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda
-(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b)
-i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6:
-(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift
-(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5)))
-(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1
-t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda
-(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T
-x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in
-(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t
-t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n
-v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2:
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind
-b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u
-(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda
-(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4)
-(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2))
-(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda
-(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2
-t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda
-(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0
-x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u
-(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift
-(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T
-(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1
-x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S
-O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i)
-v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O
-t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or
-(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i
-v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O
-x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11
-\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift
-(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda
-(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda
-(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2
-T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or
-(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0
-(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S
-O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O
-x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0)))
-(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i
-v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0
-i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x
-x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0
-(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift
-(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda
-(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1
-x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S
-O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u
-(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1:
-T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2)
-\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda
-(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3)
-w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda
-(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u
-u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda
-(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4:
-(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2:
-T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat
-Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
-(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda
-(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t:
-T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2
-T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4:
-(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5:
-T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T
-w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1
-t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat
-Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T
-(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2:
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4)
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast)
-i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
-(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i
-v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0
-x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4
-w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead
-(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
-T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4
-x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
-(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat
-Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2:
-T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast)
-i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
-(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0
-x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda
-(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0
-(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast)
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0)))
-(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0
-(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2))
-(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat
-Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2))
-(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1
-x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead
-(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1)
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2:
-T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))
-x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3))
-w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1
-t2 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/props.ma".
-
-include "LambdaDelta-1/subst1/defs.ma".
-
-theorem pr0_delta1:
- \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall
-(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead
-(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1:
-(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind
-Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind
-Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H
-t1 t2 H0 t0 H2))) w H1)))))))).
-
-theorem pr0_subst1_back:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t:
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1:
-T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda
-(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2
-T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1
-(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0
-i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda
-(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t:
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda
-(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t:
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x
-H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
-
-theorem pr0_subst1_fwd:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t:
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1:
-T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda
-(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2
-T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1
-(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0
-i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda
-(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t:
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda
-(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t:
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x
-H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
-
-theorem pr0_subst1:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall
-(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1
-v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2
-w2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1:
-T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1
-w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to
-(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))))))
-(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0
-t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2))))
-(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda
-(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2))
-(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2))
-(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2
-T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3
-(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2))
-(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0
-w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0
-w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4:
-(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2:
-T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i
-v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/defs.ma".
-
-inductive pr1: T \to (T \to Prop) \def
-| pr1_refl: \forall (t: T).(pr1 t t)
-| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3:
-T).((pr1 t2 t3) \to (pr1 t1 t3))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr1/props.ma".
-
-include "LambdaDelta-1/pr0/pr0.ma".
-
-theorem pr1_strip:
- \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0
-t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
-(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4:
-T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
-(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
-(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda
-(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda
-(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T
-(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5:
-T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda
-(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
-t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let
-H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t:
-T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)))
-(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x
-x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
-H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3
-t2 H0)))))))))) t0 t1 H))).
-
-theorem pr1_confluence:
- \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0
-t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
-(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4:
-T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
-(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
-(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2:
-T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_:
-(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda
-(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda
-(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def
-H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t))
-(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x:
-T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x
-H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda
-(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
-t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x
-x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
-H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr1/defs.ma".
-
-include "LambdaDelta-1/pr0/subst1.ma".
-
-include "LambdaDelta-1/subst1/props.ma".
-
-include "LambdaDelta-1/T/props.ma".
-
-theorem pr1_pr0:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H
-t2 (pr1_refl t2)))).
-
-theorem pr1_t:
- \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2
-t3) \to (pr1 t1 t3)))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3)))))
-(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda
-(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0
-t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2
-t5 H3)))))))))) t1 t2 H))).
-
-theorem pr1_head_1:
- \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall
-(k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t:
-T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k
-t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda
-(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda
-(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing
-(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k
-t3 t) H2))))))) u1 u2 H))))).
-
-theorem pr1_head_2:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall
-(k: K).(pr1 (THead k u t1) (THead k u t2))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u:
-T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u
-t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_:
-(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing
-(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k
-u t4) H2))))))) t1 t2 H))))).
-
-theorem pr1_comp:
- \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u:
-T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u))))))))
-\def
- \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t:
-T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall
-(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda
-(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2
-t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1
-t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t:
-T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead
-k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t
-u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k
-t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1
-H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4
-t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k
-t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp
-t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v
-w H))).
-
-theorem pr1_eta:
- \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in
-(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl)
-(TLRef O) (lift (S O) O t))) t)))))
-\def
- \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in
-(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift
-(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v
-(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H
-(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
-(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u))
-(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
-(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef
-O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O)
-u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind
-Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O)
-(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl
-(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u
-(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr
-not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O
-(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr2/props.ma".
-
-include "LambdaDelta-1/clen/getl.ma".
-
-theorem pr2_gen_ctail:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
-(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_:
-T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0
-(clen c) u t t2)))))))))
-\def
- \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c)
-(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T
-(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda
-(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
-(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind
-Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3
-t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4)
-(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t))
-(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1)))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4
-t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda
-(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let
-H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind
-(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
-(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c)))
-(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
-(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
-T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
-T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d
-(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr)
-u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e:
-C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
-T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
-T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k
-u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c
-t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3
-t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4
-H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen
-c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
-(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq
-nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_:
-nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3
-T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
-(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8:
-(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T
-u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda
-(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0
-(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind
-Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind
-Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
-t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
-Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
-t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda
-(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4
-(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1
-t2 H0))) H)))))).
-
-theorem pr2_gen_cbind:
- \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
-(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1)
-(THead (Bind b) v t2)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c
-(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead
-(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y
-t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
-(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v
-t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead
-(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1
-(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
-(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
-c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u)
-v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d
-(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S
-j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead
-(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3)
-(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead
-d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d
-| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
-v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
-(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v)
-H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def
-(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def
-(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr
-(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t)))
-(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v
-(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7:
-(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c
-(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j)))
-(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b)
-v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S
-x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal
-nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i
-(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t
-(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead
-c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t
-H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
-
-theorem pr2_gen_cflat:
- \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
-(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c
-(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2))
-(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2
-c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4
-H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
-(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
-c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u)
-v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0)))
-H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/defs.ma".
-
-include "LambdaDelta-1/getl/defs.ma".
-
-inductive pr2: C \to (T \to (T \to Prop)) \def
-| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to
-(pr2 c t1 t2))))
-| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2:
-T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1
-t)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr2/defs.ma".
-
-include "LambdaDelta-1/pr0/fwd.ma".
-
-include "LambdaDelta-1/getl/drop.ma".
-
-include "LambdaDelta-1/getl/clear.ma".
-
-theorem pr2_gen_sort:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to
-(eq T x (TSort n)))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort
-n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
-T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda
-(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0
-t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0
-t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda
-(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t:
-T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n)))
-(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl
-i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda
-(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0
-t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0))
-(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n)
-(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n))))
-t1 H4))))))))))))) c y x H0))) H)))).
-
-theorem pr2_gen_lref:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to
-(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c
-(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S
-n) O u)))))))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef
-n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
-T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead
-d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O
-u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t)
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u))))))))))
-(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1
-t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda
-(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t:
-T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0
-(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S
-n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n))
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u)))))))
-(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
-T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref
-t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda
-(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef
-n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4)
-in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda
-(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
-C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2
-(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in
-(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n))
-(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
-u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))
-(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O
-u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n))
-(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
-u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let
-H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr)
-u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T
-(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
-(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))))
-(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S
-n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8)))
-(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
-
-theorem pr2_gen_abst:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
-(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y:
-T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
-(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in
-(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1
-x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
-(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
-c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1
-(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b:
-B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4))))))
-(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda
-(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind
-Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2
-(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead
-(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1
-x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead
-(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead
-(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda
-(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
-Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind
-Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda
-(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0
-u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2
-x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2
-x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
-(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d
-u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0
-(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3:
-T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
-Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst)
-x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0
-x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead
-(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1
-x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
-(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2
-H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
-Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
-x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1
-t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t
-(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13:
-(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3)
-(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
-c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3
-(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2
-H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
-(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3
-H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9))))))))
-(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-
-theorem pr2_gen_cast:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
-(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c
-t1 x))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
-t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast)
-u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
-(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0))))))
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0
-t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind
-T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast)
-x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T
-(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead
-(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2
-H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2
-H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3:
-(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5
-\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1)
-H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast)
-x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def
-(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1)
-H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
-(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead
-(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2:
-T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
-(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2
-c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
-x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12
-(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11))
-(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda
-(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat
-Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
-(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda
-(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12
-(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11))
-(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
-x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat
-Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0
-H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11))
-(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
-t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
-(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2
-H5)))))))))))))) c y x H0))) H))))).
-
-theorem pr2_gen_csort:
- \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2)
-\to (pr0 t1 t2))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort
-n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_:
-C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind
-(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0
-t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
-t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c
-(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d
-(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr)
-u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
-
-theorem pr2_gen_appl:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
-(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
-t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
-u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3
-\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1)
-H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1
-x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
-(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl)
-x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat
-Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0
-H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2
-(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1
-x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind
-Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
-x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
-Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
-Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1))
-(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b:
-B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2
-H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
-(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b)
-y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat
-Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift
-(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0
-Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T
-t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda
-(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2
-x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
-x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
-Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
-Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
-x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0
-u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5
-H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3))))))))
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-(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
-t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T
-t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
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-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
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-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
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-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1
-x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
-T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T
-(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2:
-T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1)))
-(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t
-(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T
-(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3))))))
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-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O
-u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i
-H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11:
-(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3:
-T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t
-(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
-t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
-(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat
-Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))
-(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat
-Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr)
-u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2
-(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl)
-x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t
-H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
-x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13:
-(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1
-x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2
-(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl)
-x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1
-H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i
-H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind
-Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda
-(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2
-(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in
-(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
-(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2
-u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda
-(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
-O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead
-(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0
-i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda
-(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2
-x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
-x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
-Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1))
-(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
-H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3
-H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t
-(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
-O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr)
-x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T
-(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
-Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1))
-(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b:
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
-(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d
-(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
-x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
-O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t
-(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15:
-(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5)
-(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1)
-t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
-Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1))
-(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
-H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
-(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0)
-(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12))
-(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda
-(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2)
-t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T
-T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1:
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind
-x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl)
-(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1
-x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3:
-T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O
-x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
-(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3)
-x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t
-(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead
-(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
-i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead
-(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2:
-T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0)
-u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u
-x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
-(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3)
-x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6
-(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
-O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
-(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
-(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6
-H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda
-(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3:
-T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda
-(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
-t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
-(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0)
-i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind
-x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0)
-x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat
-Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
-u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3)
-t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
-O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
-x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T
-(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2:
-T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
-(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
-(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T
-x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift
-(S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3)
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
-x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
-x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O
-x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22
-\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
-x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
-(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4
-H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20)))))
-(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n
-i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead
-(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
-(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead
-(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
-(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
-(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19:
-(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat
-Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2)
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
-x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0
-u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S
-i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4)
-(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda
-(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u
-(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat
-Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda
-(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
-T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0
-(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat
-Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8)
-(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
-(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
-T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
-i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8))
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
-(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0)
-i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
-(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T
-(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
-(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4
-H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead
-c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2
-x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S
-O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat
-Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14))
-(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl)
-(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
-i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t
-(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17:
-(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
-x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat
-Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
-u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3)
-t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
-O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
-x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T
-(\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2:
-T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
-(u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
-(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T
-x7 (THead (Flat Appl) x8 x5))).(\lambda (H20: (subst0 (s (Bind x0) i) u (lift
-(S O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3)
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
-x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
-x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
-(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O
-x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23
-\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
-x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
-(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
-O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
-(Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
-(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u
-i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8
-H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i
-(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T
-x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s
-(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7
-(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
-Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7)
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift
-(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
-x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2)
-t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
-Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
-x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0
-u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0
-(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6
-(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7
-H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
-(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0)
-i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
-(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0)
-x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda
-(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20:
-(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s
-(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
-(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
-(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
-T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
-i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9))
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
-(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind
-x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S
-O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in
-(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
-O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
-(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T
-(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl)
-(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta
-c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i)
-(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6)
-(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22)))))
-(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n
-i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5
-x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4
-(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6))
-(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-
-theorem pr2_gen_abbr:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
-(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
-T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
-t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
-(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
-(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
-Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2))))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
-t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
-(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in
-(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
-b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
-c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
-t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
-t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1
-t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr)
-u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2)))))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr)
-x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0
-x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda
-(y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (H7: (pr0 t1
-x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
-t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
-x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
-(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda
-(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
-(pr2_free c0 u1 x0 H6) (or3_intro0 (\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
-(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead
-c0 (Bind b) u) t1 x1 H7)))))) t2 H5)) (\lambda (H_x0: (ex2 T (\lambda (y0:
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)))).(ex2_ind T (\lambda
-(y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
-t2))))) (\lambda (x2: T).(\lambda (H7: (pr0 t1 x2)).(\lambda (H8: (subst0 O
-x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
-t)))))) (ex2_ind T (\lambda (t: T).(subst0 O u1 x2 t)) (\lambda (t: T).(pr0 t
-x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
-Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))))
-(\lambda (x3: T).(\lambda (_: (subst0 O u1 x2 x3)).(\lambda (_: (pr0 x3
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
-x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
-(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda
-(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
-(pr2_free c0 u1 x0 H6) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
-(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
-(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1)) x0 H6 (pr2_delta (CHead c0 (Bind
-Abbr) x0) c0 x0 O (getl_refl Abbr c0 x0) t1 x2 H7 x1 H8))))))))
-(pr0_subst0_back x0 x2 x1 O H8 u1 H6)) t2 H5)))) H_x0)) H_x)))))) H4))
-(\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1
-t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr)
-u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))
-(\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 (lift (S
-O) O t2) H4))))) (pr0_gen_abbr u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda
-(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d
-(Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
-t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
-(THead (Bind Abbr) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0
-t3 t2)) H2 (THead (Bind Abbr) u1 t1) H4) in (or_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
-t3))))))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0:
-T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3)))
-(ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1
-y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda
-(z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (H6:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
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-c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x4: T).(\lambda (_: (subst0
-O u1 x2 x4)).(\lambda (_: (pr0 x4 x1)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0
-(Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0:
-T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z t3))))))) x3 x1 H13 (pr2_delta c0 d u i H1 u1 x0 H8 x3
-H14) (or3_intro1 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0)
-t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
-(Bind Abbr) u0) t1 x1))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x1))))
-(ex_intro2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
-(Bind Abbr) u0) t1 x1)) x0 H8 (pr2_delta (CHead c0 (Bind Abbr) x0) c0 x0 O
-(getl_refl Abbr c0 x0) t1 x2 H9 x1 H10)))))))) (pr0_subst0_back x0 x2 x1 O
-H10 u1 H8))))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t (THead
-(Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda
-(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0
-O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
-c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13
-(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_:
-T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z:
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr)
-u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl
-Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S
-i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl
-x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda
-(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t
-(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15:
-(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
-x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
-c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13
-(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
-u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T
-(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O
-(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr)
-u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1
-x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8)))))))
-H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x))))))
-H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
-(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
-(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2)
-H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i)))))))
-(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-
-theorem pr2_gen_void:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
-(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
-(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
-(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift
-(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1
-t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind
-Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
-t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
-b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u)
-t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
-c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1
-x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1)
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
-c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1
-x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1
-H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0
-(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3))))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
-t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T
-t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1
-x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
-T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T
-(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1)))
-(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t
-(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind
-Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8
-x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1
-x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead
-(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda
-(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0:
-T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0
-(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t
-(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14:
-(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b:
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11))
-(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
-t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b:
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S
-O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1
-t2 H5)))))))))))))) c y x H0))) H))))).
-
-theorem pr2_gen_lift:
- \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
-(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
-(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1
-t2))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1)
-(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e)
-\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e
-t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e:
-C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2)))
-(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h
-d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T
-t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda
-(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T
-(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
-(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1
-x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T
-t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda
-(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))
-x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5))))
-(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind
-Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
-t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
-(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def
-(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T
-(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2
-T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
-(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1
-x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h
-d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3)))
-(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def
-(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i
-(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d
-(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i))))
-(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d
-t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2:
-C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl
-i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0
-x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus
-i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind
-T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3)))
-(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h
-d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq
-T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0
-x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
-nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T
-(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
-H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0
-t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h
-(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2
-T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
-(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11
-H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1
-t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t
-(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
-(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
-(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13:
-(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d
-t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h)
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1
-H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1
-t2 h d H6)))))))))))))))) c y x H0))) H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr2/defs.ma".
-
-include "LambdaDelta-1/pr0/pr0.ma".
-
-include "LambdaDelta-1/getl/props.ma".
-
-theorem pr2_confluence__pr2_free_free:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0
-t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0
-t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t))
-(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2
-x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
-(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1)))))
-(pr0_confluence t0 t2 H0 t1 H))))))).
-
-theorem pr2_confluence__pr2_free_delta:
- \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall
-(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2)
-\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t))))))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0
-t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0
-t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4
-t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
-(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4:
-(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
-(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2
-c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2
-x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2:
-T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
-(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7:
-(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0
-H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u))))))
-(pr0_confluence t0 t4 H1 t1 H))))))))))))).
-
-theorem pr2_confluence__pr2_delta_delta:
- \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall
-(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u:
-T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d
-(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to
-(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t))))))))))))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u:
-T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i
-c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0
-i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda
-(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t:
-T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4
-x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0
-t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2
-x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
-w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
-(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
-(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda
-(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0
-u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
-(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x
-x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))
-x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8))
-(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T
-(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind
-T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T
-(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0:
-T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0
-t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
-w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
-(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
-(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H
-t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2))
-(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2
-w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2
-x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t:
-T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i
-i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0
-i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15:
-(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d
-u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9
-(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def
-(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15
-\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0)))
-H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0:
-C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind
-Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u)
-(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0
-(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono
-c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda
-(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x
-x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c
-(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0
-(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind
-(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t:
-T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda
-(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1
-x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in
-(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0
-(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T
-(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0
-t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i
-u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
-(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i
-u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c
-t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2
-x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T
-(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1
-x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u
-x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11)))
-(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10))
-(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x
-H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3
-H0))))))))))))))))))).
-
-theorem pr2_confluence:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
-(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
-t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in
-pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_:
-(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T
-(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with
-[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3:
-(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T
-t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c
-t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind
-T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5:
-T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4
-t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5:
-T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0
-t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t:
-T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0)
-\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6:
-T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda
-(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6
-t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5
-t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6
-t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7:
-T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t:
-T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7:
-T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0
-t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2
-H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) |
-(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1
-c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c
-(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d
-(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda
-(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14:
-(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c
-(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T
-(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))))
-(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d
-(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda
-(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16:
-(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda
-(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i
-H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1
-(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c)
-(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T
-t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4
-H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3
-t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0)
-\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4)
-\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda
-(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0
-(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1
-t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind
-T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4)
-\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda
-(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr)
-u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12
-\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6:
-T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6
-t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2
-t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C
-c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c
-(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda
-(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7
-t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
-t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0
-t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
-t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1)
-(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6
-(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14
-H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow
-(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T
-t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to
-((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0
-t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
-t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T
-t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to
-((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda
-(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2
-(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to
-((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda
-(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr)
-u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6
-t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10
-H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1
-(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c)
-(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T
-t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C
-c) (refl_equal T t0) (refl_equal T t1)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr2/defs.ma".
-
-include "LambdaDelta-1/pr0/props.ma".
-
-include "LambdaDelta-1/getl/drop.ma".
-
-include "LambdaDelta-1/getl/clear.ma".
-
-theorem pr2_thin_dx:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u
-t2)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0)))))
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0
-t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u
-(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0
-t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i
-H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0
-t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2
-u)))))))))))) c t1 t2 H)))))).
-
-theorem pr2_head_1:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
-(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
-u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0:
-T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0:
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0
-(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2
-t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1
-t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c
-u1 u2 H)))))).
-
-theorem pr2_head_2:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
-(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u
-t2)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u)
-(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead
-k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c
-(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
-(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k
-u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3
-t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d:
-C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0))
-\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t:
-T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u
-t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0
-(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4)
-\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u))
-\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1:
-(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4
-t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0
-(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b)
-u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
-c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c
-(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind
-Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
-(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
-(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d
-(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr)
-u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in
-(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T
-u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda
-(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c
-(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u)
-t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1:
-(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
-T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
-c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u
-t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
-(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b)
-u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
-Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
-(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
-(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
-(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u
-t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n)
-(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3)
-(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead
-(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u)))))))))))))
-i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
-T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind
-Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall
-(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c
-(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0
-(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2:
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4:
-(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1:
-C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in
-(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c)
-(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f)
-u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4)
-(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t)
-(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1:
-(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
-T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
-c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u
-t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
-(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f)
-u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
-Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
-(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
-(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
-(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u
-t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n)
-(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3)
-(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead
-(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u)))))))))))))
-i)))))) k) y t1 t2 H0))) H)))))).
-
-theorem clear_pr2_trans:
- \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
-(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
-\def
- \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1
-t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1:
-C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_:
-(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1:
-C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c
-(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
-
-theorem pr2_cflat:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free
-(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
-(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u
-i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c
-t1 t2 H)))))).
-
-theorem pr2_ctail:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0)
-t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
-(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail
-Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
-
-theorem pr2_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
-T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to
-(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
-(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind
-b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda
-(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2))
-(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2
-(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
-(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b)
-v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3:
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5:
-(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1:
-C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in
-(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind
-Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))
-(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr)
-u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
-v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind
-b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind
-Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
-(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d
-(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr)
-u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in
-(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind
-T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def
-(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B
-Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match
-(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c
-(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda
-(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr)
-u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda
-(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda
-(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0)
-(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
-(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4)))))))))))))
-y t1 t2 H1))) H0)))))))).
-
-theorem pr2_lift:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
-d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift
-h d t1) (lift h d t2)))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1
-t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c
-(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e)
-\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0
-e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d)))))))
-(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4
-t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1:
-C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c
-(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def
-(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0
-(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
-O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
-(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c
-(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9:
-(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11:
-(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i)
-(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i
-H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u
-H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h
-(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c
-(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x
-(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S
-i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead
-x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d
-t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7
-h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h)
-(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3)
-(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h
-H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr2/defs.ma".
-
-include "LambdaDelta-1/pr0/subst1.ma".
-
-include "LambdaDelta-1/pr0/fwd.ma".
-
-include "LambdaDelta-1/csubst1/getl.ma".
-
-include "LambdaDelta-1/csubst1/fwd.ma".
-
-include "LambdaDelta-1/subst1/subst1.ma".
-
-include "LambdaDelta-1/getl/drop.ma".
-
-theorem pr2_delta1:
- \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2)
-\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2
-t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0)
-(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2
-H0 t0 H2))) t H1)))))))))).
-
-theorem pr2_subst1:
- \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
-\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c
-w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1
-t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T
-(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))
-(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1:
-T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda
-(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
-(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1:
-T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T
-(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))))
-(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))
-(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))
-(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4
-x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v
-t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v
-(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr)
-u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
-(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda
-(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda
-(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c
-(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2:
-T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda
-(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda
-(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1
-x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2:
-T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not
-(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0:
-T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
-T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t
-x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c
-w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x
-H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t
-H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let
-H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in
-(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind
-Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v)
-(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead
-e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal
-C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v)
-(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d
-(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _
-t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u)
-(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in
-(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl
-i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u
-(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C
-d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in
-(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x
-t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t
-w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21:
-(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
-T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20))))
-(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9)))))))
-H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0
-H5))))))))))))))) y t1 t2 H1))) H0)))))))).
-
-theorem pr2_gen_cabbr:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
-\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
-a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
-(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
-x1 x2))))))))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e:
-C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to
-(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0
-a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda
-(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1
-x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
-nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
-C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
-d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d
-x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2:
-T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
-x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0
-(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda
-(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T
-(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
-x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda
-(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t))
-H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4
-(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0
-H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S
-O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e:
-C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
-(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
-a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1:
-T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda
-(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2))
-(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2:
-T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1)
-x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x
-(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2:
-T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2)))
-(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10:
-(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0))
-H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1
-d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12:
-(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0:
-T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0
-t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2:
-T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O)
-d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d
-(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3:
-T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3)))
-(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr)
-u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i)
-(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0
-(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr)
-d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda
-(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_:
-C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2:
-C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0
-t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4:
-T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr)
-x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1
-(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl
-i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0
-(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i))))
-(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0
-u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6:
-T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i))
-x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop
-(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0:
-T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6)
-H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i))
-x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S
-i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0
-x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28:
-(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda
-(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind
-nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S
-O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in
-(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S
-i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9:
-T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9)))
-(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S
-i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8
-(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift
-(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat
-(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S
-i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let
-H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n:
-nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12))
-in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10)))
-(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O)
-d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32))))))))
-(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30))))))
-(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S
-i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i
-H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12
-c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i
-(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12))))
-(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n:
-nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def
-(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15
-\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in
-(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind
-Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2:
-T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let
-H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1))
-H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead
-e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _
-_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)
-(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in
-((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind
-Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d
-e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind
-Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0:
-T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r
-T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u
-(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2)))
-(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1:
-C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda
-(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0)
-t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2:
-T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t
-x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind
-T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0)
-(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i)
-(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i
-(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i
-x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10))))))
-(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0)
-H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T
-(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S
-O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2)))
-(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0
-u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T
-(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1
-(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O)
-d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq
-T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0
-x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13
-(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t
-(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u
-(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0
-(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d
-(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n:
-nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3
-H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S
-O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S
-O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift
-(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12))))))))))
-(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0
-x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr2/defs.ma".
-
-inductive pr3 (c: C): T \to (T \to Prop) \def
-| pr3_refl: \forall (t: T).(pr3 c t t)
-| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3:
-T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/props.ma".
-
-include "LambdaDelta-1/pr2/fwd.ma".
-
-theorem pr3_gen_sort:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to
-(eq T x (TSort n)))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort
-n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t:
-T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda
-(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t:
-T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2:
-T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
-(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3
-t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda
-(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t:
-T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n))
-\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T
-(TSort n)))) t1 H4))))))))) y x H0))) H)))).
-
-theorem pr3_gen_abst:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
-(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
-T).(pr3 (CHead c (Bind b) u) t1 t2))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y:
-T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead
-(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0:
-T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c
-(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t
-(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda
-(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0
-x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl
-(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda
-(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda
-(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1))
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst)
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
-(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
-t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1
-t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
-(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
-c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
-t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
-(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0
-x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4:
-T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5
-t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3
-(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
-(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst)
-x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))
-x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u:
-T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u))))))))))
-H11)))))))) H6)))))))))))) y x H0))))) H))))).
-
-theorem pr3_gen_cast:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
-(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c
-t1 x))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1
-t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T
-t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall
-(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0
-t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall
-(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0
-x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
-x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c
-x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T
-(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1)))))
-(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
-T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1:
-T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
-t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4:
-(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t:
-T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def
-(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c
-x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
-Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5:
-T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2
-c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda
-(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5))
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
-Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast)
-x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2
-x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
-t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
-t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T
-t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16:
-(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat
-Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead
-(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5
-(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15)
-(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3
-t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c
-x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4
-H2))) H6)))))))))))) y x H0))))) H))))).
-
-theorem pr3_gen_lift:
- \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
-(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
-(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1
-t2))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1)
-(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e)
-\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e
-t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda
-(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T
-(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2)))))))
-(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h
-d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T
-t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t:
-T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e:
-C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h
-d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda
-(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
-T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2
-(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5:
-T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda
-(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5:
-(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1
-(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in
-(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e
-x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5:
-T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d
-x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift
-h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4
-(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda
-(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T
-(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2
-H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x
-H0)))) H)))))).
-
-theorem pr3_gen_lref:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to
-(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T x (lift (S n) O v))))))))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef
-n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t:
-T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y
-x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or
-(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T
-t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u:
-T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
-C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t))))
-(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3:
-T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or
-(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef
-n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4)
-in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind
-Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
-v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef
-n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr)
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T
-t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef
-n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or
-(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def
-(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8
-(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d:
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n))
-(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead
-d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
-v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0
-(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def
-(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C
-T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind
-Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
-v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda
-(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift
-c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda
-(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq
-T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T
-t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3
-(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl
-n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v:
-T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3
-(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u:
-T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
-C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13)))))
-H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))).
-
-theorem pr3_gen_void:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
-(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1
-(lift (S O) O x)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c
-(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y
-x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1
-(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c
-(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t
-(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1
-(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind
-Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1
-(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c
-(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1))
-(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b)
-u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c
-t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall
-(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
-(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let
-H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0
-x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O
-t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind
-Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void)
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
-(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
-c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq
-T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O)
-O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind
-Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind
-T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind
-Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2)))
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O
-t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3
-(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c
-(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c
-(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void)
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda
-(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4
-x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c
-(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14
-(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead
-c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13:
-(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
-(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind
-Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift
-(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7:
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O
-t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
-c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))
-(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S
-O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void)
-O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
-
-theorem pr3_gen_abbr:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
-(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr)
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
-O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda
-(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O)
-O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind
-Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3
-(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t:
-T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind
-Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda
-(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0
-x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0)
-x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0
-x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1)))))
-(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
-T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1:
-T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
-(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
-t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1
-t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2
-(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u:
-T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda
-(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z:
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0)
-z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1
-(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
-b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead
-c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z
-t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u:
-T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda
-(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z:
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0)
-z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2
-c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c
-(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2
-(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_:
-T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z:
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0)
-z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c
-(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3))))
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr)
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c
-(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind
-T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind
-Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3
-(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2
-x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3)))
-in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind
-Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c
-(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3
-t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c
-(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c
-x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T
-(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
-(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4
-x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1
-(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4
-H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O
-t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr)
-x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O)
-O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u:
-T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1
-x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c
-(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4:
-T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4)
-x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall
-(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S
-O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3
-(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S
-O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda
-(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5
-x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr)
-x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0)
-x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
-T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5
-x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0)
-(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3
-x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c
-(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1
-x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O
-t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T
-(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0)))
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
-(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq
-T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4:
-T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1
-x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0)
-x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall
-(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S
-O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3
-(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S
-O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda
-(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6
-x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr)
-x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0)
-x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
-T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
-O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
-(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6
-x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1
-H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind
-Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7
-(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17))
-(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O
-t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr)
-x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4
-(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing
-(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3
-(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10))))))
-H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing
-(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4)
-(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c
-(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
-
-theorem pr3_gen_appl:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
-(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
-c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1
-t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1
-(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda
-(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall
-(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))))
-(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t
-(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda
-(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))
-(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat
-Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
-c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
-(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0
-x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda
-(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2
-t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat
-Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
-z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
-(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
-t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1
-t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl)
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1
-t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2
-x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11
-\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t
-(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
-T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
-(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t:
-T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3
-(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
-t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4
-x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T
-(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
-T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4
-x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl)
-x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c
-x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T
-(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10
-x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15:
-(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda
-(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T
-T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro
-T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5:
-T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1
-t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10
-(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
-(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8:
-T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3
-c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9
-(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2
-x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9)
-x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))
-x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17
-(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7))
-(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
-Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
-Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq
-T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4
-x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2
-x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
-t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c
-(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12
-\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t
-(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
-T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
-(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t:
-T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
-Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c
-(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind
-Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u)
-x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl)
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6:
-T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T
-x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7
-(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0
-x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7)
-x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2
-(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl)
-x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
-(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2)))
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
-(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let
-H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7
-(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
-(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
-Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
-t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4))
-H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2)
-x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
-
-theorem pr3_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1:
-T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind
-b) u1) t1 (lift (S O) O x)))))))))
-\def
- \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
-(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind
-b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3
-(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B
-Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x:
-T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def
-(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr)
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1)
-t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
-Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1)
-t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x
-(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3
-(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1
-H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
-O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H:
-(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
-T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1
-\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c
-(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_:
-(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
-T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1
-\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c
-(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void)
-u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda
-(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void)
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
-u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead
-c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0
-x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall
-(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1
-H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1
-(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/fwd.ma".
-
-include "LambdaDelta-1/iso/props.ma".
-
-include "LambdaDelta-1/tlist/props.ma".
-
-theorem pr3_iso_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat
-Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w))
-u2))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
-(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall
-(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
-(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2:
-T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to
-(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq
-T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda
-(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso
-(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T
-(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef
-i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T
-(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind
-Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O
-v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0
-(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift
-(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t)
-\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S
-i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C
-(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind
-Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1)
-H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d
-(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead
-x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind
-Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1
-(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13
-\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def
-(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d
-H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d
-H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15)))))))
-H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0:
-TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef
-i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P:
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w))
-u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3
-\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda
-(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t
-x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def
-(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
-t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1)
-H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead
-(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t
-x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1)))
-u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
-w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t
-x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
-Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c
-(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t
-(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
-(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead
-(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P:
-Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead
-(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1))
-(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst)
-x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1
-(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t
-x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5))))))))))
-H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
-(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat
-Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead
-(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda
-(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0)
-x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind
-x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c
-(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1
-x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0
-(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P:
-Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat
-Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift
-(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2))
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0
-H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind
-x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1
-(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead
-(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift
-(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O
-(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2
-(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2))))
-u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12
-c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat
-Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10
-(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
-
-theorem pr3_iso_appls_cast:
- \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1
-\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2:
-T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c
-(THeads (Flat Appl) vs t) u2))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
-TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0
-(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2))))))
-(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda
-(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1
-\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
-t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0
-x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def
-(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall
-(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat
-Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat
-Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2:
-(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H:
-((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2)
-\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall
-(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2:
-T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead
-(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P:
-Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead
-(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda
-(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat
-Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
-Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2)
-(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2
-(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2)
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl)
-x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl)
-t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2:
-T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v
-t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in
-(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat
-Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat
-Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl)
-t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3))
-(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1
-t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))
-(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c
-(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6:
-(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0
-x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b)
-u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1))
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads
-(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6
-(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead
-(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast
-Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c
-(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1)
-(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind
-Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2
-(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c
-t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3:
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b)
-y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
-(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat
-Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda
-(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0
-Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t))
-(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead
-(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda
-(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat
-Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat
-Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t))
-c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0
-(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead
-(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads
-(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda
-(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0
-Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2
-c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead
-(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead
-(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0)
-x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2
-(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4)
-x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift
-(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead
-c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind
-x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7)
-(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift
-(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
-O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
-(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2)
-(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5)
-x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
-
-theorem pr3_iso_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2:
-T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t))
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O v1) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
-(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c
-(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead
-(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let
-H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T
-T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1
-t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst))))))))
-(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1
-z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq
-T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_:
-(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0:
-T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P:
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl)
-x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S
-O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl))
-(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead
-(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1
-t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u:
-T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr)
-x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b)
-v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall
-(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3)
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def
-(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in
-(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
-x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))))
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind
-Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
-T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda
-(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4
-x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2)
-t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return
-(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow
-Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst)
-x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0
-| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind
-Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0]))
-(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T
-x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda
-(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def
-(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def
-(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst
-H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H
-Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22
-\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_:
-False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t))
-(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13)))))))
-H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind
-Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
-x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift
-(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead
-(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O
-v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop
-(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst)
-x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead
-(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead
-(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2
-x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst)
-x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2
-(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead
-(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta
-x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3)
-(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead
-(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1)))
-(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4)))))))))
-H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3:
-T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O)
-O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0)
-y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
-v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0
-Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1
-x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
-O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1
-x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t
-(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind
-b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat
-Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl)
-(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq
-T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2
-x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal
-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match
-k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
-((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0
-_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
-((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
-(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def
-(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17)
-in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18)
-in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2
-x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0
-Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat
-Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8)
-(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S
-O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O
-x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c
-(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3
-(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3
-(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11:
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
-(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1)
-t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat
-Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b)
-(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift
-(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2)
-v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O
-(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c
-(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
-x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O
-x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind
-x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl)
-x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2))
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0
-H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5
-H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl)
-(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S
-O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
-(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O))))
-H10))) u2 H6))))))))))))) H3)) H2)))))))))).
-
-theorem pr3_iso_appls_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u:
-T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs
-(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2:
-T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c
-(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
-t))) u2)))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda
-(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
-TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
-b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead
-(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c:
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b)
-u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t))
-u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1)))))
-(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall
-(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
-t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
-b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda
-(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda
-(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v
-(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def
-(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
-b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead
-(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T
-T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
-Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))))
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat
-Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat
-Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl)
-t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P:
-Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl)
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8
-(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
-t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat
-Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T
-T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead
-(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3)
-u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1
-(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0
-x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind
-b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0
-(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
-t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat
-Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S
-O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u
-(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c
-(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead
-(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P:
-Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t)
-t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl)
-t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead
-(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta
-x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr)
-x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3
-(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
-t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
-Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda
-(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u
-(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead
-(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl)
-t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t
-(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c
-(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2)
-H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
-(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P:
-Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1
-H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0)
-x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind
-x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat
-Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl)
-(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl
-x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat
-Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2)
-(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4)
-(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c
-(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind
-x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5
-(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat
-Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat
-Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3)
-(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2
-H7)))))))))))))) H4)) H3))))))))) vs)))))).
-
-theorem pr3_iso_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u:
-T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t))
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl)
-(lifts (S O) O vs) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs:
-TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0:
-T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c:
-C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P:
-Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t)
-t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda
-(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso
-(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts:
-TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0:
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead
-(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0))
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat
-Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0:
-T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl)
-(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat
-Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P:
-Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t))
-(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0))
-u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl)
-(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let
-H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0))
-(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl)
-t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0)
-ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind
-b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2
-(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0)))
-(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda
-(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall
-(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to
-((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P:
-Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O
-t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t
-(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat
-Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c
-(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl)
-(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1:
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead
-(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0
-t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads
-(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c
-(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0)))
-u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b
-H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_:
-((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3
-c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads
-(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to
-(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2))
-u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
-(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead
-(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift
-(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2:
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1
-ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1
-ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0
-(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2))
-u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead
-(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to
-(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2
-(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8:
-(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads
-(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O
-t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
-(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads
-(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat
-Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t))
-(lifts_tapp (S O) O t ts))))))))))) vs))).
-
-theorem pr3_iso_beta:
- \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat
-Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c
-u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind
-Abbr) v t) u2))))))))
-\def
- \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2:
-T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t))
-u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2)
-\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind
-Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2))))
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
-b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c
-(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst)
-w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v
-x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T
-u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0)
-\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T
-(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t)
-t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead
-(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2:
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
-b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t)
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v
-x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0
-x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b)
-u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5)
-in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
-x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
-(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0]))
-(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0)
-\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in
-(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0:
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1
-H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0
-H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c
-(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2)
-(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2))
-(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind
-Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5
-(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v
-x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2
-x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in
-(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1
-x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
-(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead
-(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead
-(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
-\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2)
-(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 |
-(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0)
-x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1
-x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda
-(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0))))
-H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0))
-H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind
-b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b:
-B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2))
-H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b
-Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in
-False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with [])
-in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
-
-theorem pr3_iso_appls_beta:
- \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1
-\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in
-(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr)
-v t)) u2)))))))))
-\def
- \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall
-(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl)
-v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1
-u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat
-Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w:
-T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P:
-Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda
-(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1:
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead
-(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl)
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P:
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))
-u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c:
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1:
-(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v
-(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def
-(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
-Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl)
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2)
-(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))
-(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)))
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat
-Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P:
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl)
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0
-(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead
-(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat
-Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T
-T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat
-Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c
-(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3
-c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))
-(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u:
-T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1)
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
-(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1
-c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0
-x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1
-(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1)
-(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr)
-t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead
-(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2
-(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t
-x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3:
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst)
-w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead
-(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq
-B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v
-(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda
-(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr)
-v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2))
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
-(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c
-(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead
-(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda
-(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst)
-w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O)
-O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead
-(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl)
-(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1)
-x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O
-x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl)
-(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12
-(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c
-(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7)
-(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift
-(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
-O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
-(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2)
-(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5)
-x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/defs.ma".
-
-include "LambdaDelta-1/pr1/defs.ma".
-
-theorem pr3_pr1:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1
-t2))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t:
-T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0
-t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c:
-C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/props.ma".
-
-include "LambdaDelta-1/pr2/pr2.ma".
-
-theorem pr3_strip:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
-(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
-T).(pr3 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
-t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t
-t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3
-t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t
-t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2
-t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda
-(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2
-t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t:
-T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda
-(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t:
-T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c
-t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2
-x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t))
-(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda
-(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T
-(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c
-x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0))))))))))
-t0 t1 H)))).
-
-theorem pr3_confluence:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
-(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
-T).(pr3 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
-t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t
-t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3
-t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t
-t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2
-t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda
-(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda
-(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4
-t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c
-t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2
-t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)))
-(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2
-x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t))
-(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda
-(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T
-(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5
-c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1
-H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/pr1.ma".
-
-include "LambdaDelta-1/pr2/props.ma".
-
-include "LambdaDelta-1/pr1/props.ma".
-
-theorem clear_pr3_trans:
- \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to
-(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
-\def
- \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t:
-T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5:
-T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3
-t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
-
-theorem pr3_pr2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
-
-theorem pr3_t:
- \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall
-(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0
-t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3
-c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall
-(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3:
-(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
-
-theorem pr3_thin_dx:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
-(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u
-t2)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0:
-T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t:
-T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0
-t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u
-t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c
-t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
-
-theorem pr3_head_1:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
-(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
-u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall
-(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda
-(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2:
-T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
-(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c
-(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing
-c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t)
-(H2 k t)))))))))) u1 u2 H)))).
-
-theorem pr3_head_2:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
-(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u
-t2)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u)
-(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0))))
-(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_:
-(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u
-t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0)
-(THead k u t4) H2))))))) t1 t2 H)))))).
-
-theorem pr3_head_21:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
-(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
-(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c
-u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
-
-theorem pr3_head_12:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
-(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
-(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c
-u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
-
-theorem pr3_cflat:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
-(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v:
-T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f:
-F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
-(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead
-c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c
-(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
-
-theorem pr3_flat:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
-(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead
-(Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
-u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
-(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f
-u2))))))))).
-
-theorem pr3_pr0_pr2_t:
- \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c:
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
-(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
-t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
-(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_:
-(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k
-u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4:
-(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead
-c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d
-(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t))))
-(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8:
-(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d
-(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda
-(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind
-Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2
-(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11
-\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
-with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2)
-(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b)
-u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
-u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind
-b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr
-b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0:
-T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3
-(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4
-t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t)
-(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x
-t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr)
-u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind
-Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17))))))
-(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f:
-F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr)
-u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O
-(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c
-(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind
-Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda
-(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4
-t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k
-u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind
-(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to
-((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t)
-\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda
-(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr)
-u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))
-\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2
-(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0)
-(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
-(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f:
-F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr)
-u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))
-\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2
-(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0)
-(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f
-u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
-
-theorem pr3_pr2_pr2_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
-u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
-T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3
-(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
-K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0
-t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
-(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
-(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1:
-C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y:
-C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
-T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4
-t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
-t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5
-(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
-C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
-(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
-t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
-c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
-(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n:
-nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
-t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t)
-(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
-(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3
-(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
-(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0
-| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind
-b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14
-\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
-with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t)
-(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
-Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr)
-u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let
-H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in
-(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind
-T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i
-O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda
-(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let
-H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O
-i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n
-u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7))
-(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda
-(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0
-x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind
-Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead
-c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i)
-(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1)
-(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x
-O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14))
-H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0
-(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6
-(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0
-(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5
-H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0)
-H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0
-(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4
-t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr)
-u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl
-(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1)
-t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t)
-(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta
-(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind
-Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4
-t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat
-f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6
-(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f)
-c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10)))))
-i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))).
-
-theorem pr3_pr2_pr3_t:
- \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
-(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to
-(pr3 (CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
-(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3
-(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
-u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
-(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2)
-\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1
-u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
-u1 H3)))))))))) t1 t2 H)))))).
-
-theorem pr3_pr3_pr3_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
-u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
-(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1
-t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
-K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
-(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
-t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3
-(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0:
-T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0
-t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
-
-theorem pr3_lift:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
-d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift
-h d t1) (lift h d t2)))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1
-t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h
-d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda
-(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0
-t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d
-t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2
-H0)))))))).
-
-theorem pr3_eta:
- \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind
-Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v
-(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
-\def
- \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind
-Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead
-(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c
-(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w
-u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead
-(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat
-Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u
-(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat
-Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)))
-(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S
-O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing
-(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O)
-(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O))
-(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))
-(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind
-Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u
-(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead
-(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/defs.ma".
-
-include "LambdaDelta-1/pr2/subst1.ma".
-
-theorem pr3_subst1:
- \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2)
-\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c
-w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0:
-T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1
-w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1:
-T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1
-w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5:
-T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v
-t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i
-v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T
-(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T
-(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2)))
-(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3
-x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5
-w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5
-w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v
-t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1
-i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c
-e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
-
-theorem pr3_gen_cabbr:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
-(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
-\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
-a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
-(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a
-x1 x2))))))))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u:
-T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
-C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
-(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1
-d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)))))))))))))))
-(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
-(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_:
-(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0
-a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d
-x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2)))
-(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4:
-T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u:
-T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
-C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
-(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2:
-T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1
-x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
-(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4:
-(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0
-a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d
-x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda
-(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
-x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1
-d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda
-(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2))
-(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2:
-T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O)
-d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d
-u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x
-x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0
-e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/props.ma".
-
-include "LambdaDelta-1/wcpr0/getl.ma".
-
-theorem pr3_wcpr0_t:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1:
-T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0
-t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3:
-C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2:
-T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1)
-(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t:
-T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
-(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead
-c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3
-(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0))
-(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7:
-(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq
-C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c:
-C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_:
-(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0
-k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr)
-u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda
-(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k
-u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind
-Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2:
-C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3:
-T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr)
-x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T
-(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3
-(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6
-x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta
-(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t
-(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1
-H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3
-H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7)))
-H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2
-H2)))))))))))))) c2 c1 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Base-1/theory.ma".
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-definition r:
- K \to (nat \to nat)
-\def
- \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i |
-(Flat _) \Rightarrow (S i)])).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/r/defs.ma".
-
-include "LambdaDelta-1/s/defs.ma".
-
-theorem r_S:
- \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
-i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
-(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
-f) i))))) k).
-
-theorem r_plus:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
-(plus (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
-(Flat f) i) j))))) k).
-
-theorem r_plus_sym:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
-(plus i (r k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
-F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
-
-theorem r_minus:
- \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
-(minus (r k i) (S n)) (r k (minus i (S n)))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
-K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
-n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
-F).(minus_x_Sy i n H)) k)))).
-
-theorem r_dis:
- \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
-\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i:
-nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i)
-(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H:
-((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_:
-((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i:
-nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
-((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
-((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
-(i: nat).(refl_equal nat (S i)))))))) k).
-
-theorem s_r:
- \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0
-i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i))))
-(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
-
-theorem r_arith0:
- \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
-\def
- \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
-nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
-nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
-(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
-
-theorem r_arith1:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
-i)) (S j)) (minus (r k i) j))))
-\def
- \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
-(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
-(minus (r k i) j)) (r k (S i)) (r_S k i)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-definition s:
- K \to (nat \to nat)
-\def
- \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) |
-(Flat _) \Rightarrow i])).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/s/defs.ma".
-
-theorem s_S:
- \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S
-i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s
-(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat
-f) i))))) k).
-
-theorem s_plus:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
-(plus (s k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j)))))
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s
-(Flat f) i) j))))) k).
-
-theorem s_plus_sym:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
-(plus i (s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda
-(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq
-nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j))
-(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j:
-nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
-
-theorem s_minus:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s
-k (minus i j)) (minus (s k i) j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j))))))
-(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j
-i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i)
-j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H))))))
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j
-i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
-
-theorem minus_s_s:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s
-k j)) (minus i j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_:
-B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j)))))
-(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i
-j))))) k).
-
-theorem s_le:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i)
-(s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
-nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_:
-F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
-
-theorem s_lt:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i)
-(s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
-nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda
-(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
-
-theorem s_inj:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j))
-\to (eq nat i j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b:
-B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s
-(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda
-(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
-
-theorem s_inc:
- \forall (k: K).(\forall (i: nat).(le i (s k i)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i))))
-(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s
-(Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n
-(s (Flat f) i)))) k).
-
-theorem s_arith0:
- \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
-\def
- \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n:
-nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal
-nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
-
-theorem s_arith1:
- \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
-\def
- \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n
-i)) (refl_equal nat i) (minus i O) (minus_n_O i))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubc/arity.ma".
-
-include "LambdaDelta-1/csubc/getl.ma".
-
-include "LambdaDelta-1/csubc/drop1.ma".
-
-include "LambdaDelta-1/csubc/props.ma".
-
-theorem sc3_arity_csubc:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
-t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall
-(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
-A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
-C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c:
-C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_:
-(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T
-(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0)))
-(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2
-n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n
-is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
-A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall
-(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
-a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda
-(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let
-H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in
-(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2:
-C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u))))
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1
-(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr)
-(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x
-(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def
-H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2:
-C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2
-(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2
-x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u))
-x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind
-Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0
-(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
-c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
-(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
-A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda
-(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2)))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))))
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq
-C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc
-g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1
-(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is
-(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr)
-(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind
-C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr)
-(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in
-(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y
-(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O
-u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i)
-O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O)
-(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4)
-(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans
-is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef
-i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_:
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda
-(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
-is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
-w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq
-K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
-A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is
-(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13:
-(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr)
-x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1
-(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0
-(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14)
-in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return
-(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13)
-in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12))
-(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b:
-B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2)))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda
-(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq
-K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_:
-(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is
-i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind
-Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat
-_) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 (lift1
-is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) H5))))))))))))))))
-(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1:
-(arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: C).(\forall (is:
-PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
-(asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is:
-PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g
-d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 H3 Abst d
-u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is
-i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1
-(ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x:
-C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is i)
-d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def
-(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is
-i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans
-is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans
-is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda
-(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst)
-(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1
-(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C
-(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u))))
-(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
-is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
-w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C
-x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_:
-T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2
-C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u))))
-(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0
-(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
-c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C
-x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x
-x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0))
-H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def
-(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0:
-T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef
-(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1
-t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0
-H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1
-(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is
-i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_:
-T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
-is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
-w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq
-K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
-A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
-T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
-C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is
-(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_:
-(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr)
-x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x
-(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def
-(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind
-Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef
-(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2
-(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let
-H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in
-(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S
-(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g
-x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0)
-H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13))
-(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2:
-T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3:
-C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b:
-B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2)))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda
-(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq
-K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_:
-(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is
-i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind
-Abst) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
-_) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 (lift1
-is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) H6)))))))))))))))))
-(\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2:
-((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
-C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0
-a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1
-(CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2
-(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H5:
-(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g d1 c2)).(let H_y
-\def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead (Bind b) (lift1 is u)
-(lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u)
-(lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) (Ss is)
-(drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) (csubc_head
-g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is (THead
-(Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g
-a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
-\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is
-u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c
-(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is:
-PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g
-d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is:
-PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g
-d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))
-(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall
-(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g
-a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead
-(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d:
-C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d
-c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1
-is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind
-Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1
-is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2
-(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst)
-(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d:
-C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0:
-PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1
-is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3
-g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8
-d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr
-(\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) a1 a2 TNil) in (H_y d w
-(lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def (csubc_drop1_conf_rev g is0
-d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C (\lambda (c3: C).(drop1 is0
-c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w)
-(lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: C).(\lambda (H10: (drop1
-is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T (lift1 (papp (Ss is0)
-(Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind Abbr) w) t1))
-(eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g a2 (CHead d
-(Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 (papp is0 is)
-u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x u (drop1_trans
-is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x d H11 (lift1
-(papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 H10 is c H4) x
-(csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 is)) (lift1 (Ss
-is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) t0))))) H9))) H6)) H6
-(lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) is0 d (lift1 is u) (H1
-d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss
-is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 (Ss is) t0))))))))) (lift1
-is (THead (Bind Abst) u t0)) (lift1_bind Abst is u t0))))))))))))))))
-(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u
-a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
-\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is
-u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0
-(AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: PList).((drop1
-is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2
-(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4:
-(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y
-\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6
-\def H_y0 in (land_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d:
-C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d
-c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0)))))))))
-(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2
-(lift1 is t0) (AHead a1 a2))).(\lambda (H8: ((\forall (d: C).(\forall (w:
-T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2
-d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))))).(let H_y1 \def (H8
-c2 (lift1 is u) H_y PNil) in (eq_ind_r T (THead (Flat Appl) (lift1 is u)
-(lift1 is t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2))
-(lift1 is (THead (Flat Appl) u t0)) (lift1_flat Appl is u t0)))))
-H6)))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
-A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (d1:
-C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1
-c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda
-(_: (arity g c t0 a0)).(\lambda (H3: ((\forall (d1: C).(\forall (is:
-PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a0
-c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4:
-(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y
-\def (sc3_cast g a0 TNil) in (eq_ind_r T (THead (Flat Cast) (lift1 is u)
-(lift1 is t0)) (\lambda (t1: T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1
-is H4 c2 H5) (lift1 is t0) (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast)
-u t0)) (lift1_flat Cast is u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0:
-T).(\lambda (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall
-(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g
-d1 c2) \to (sc3 g a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2:
-(leq g a1 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is
-d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2
-(lift1 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
-
-theorem sc3_arity:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to (sc3 g a c t)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y
-(drop1_nil c) c (csubc_refl g c))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sn3/defs.ma".
-
-include "LambdaDelta-1/arity/defs.ma".
-
-include "LambdaDelta-1/drop1/defs.ma".
-
-definition sc3:
- G \to (A \to (C \to (T \to Prop)))
-\def
- let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c:
-C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t
-(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead
-a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is
-t)))))))))]))) in sc3.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sc3/defs.ma".
-
-include "LambdaDelta-1/sn3/lift1.ma".
-
-include "LambdaDelta-1/nf2/lift1.ma".
-
-include "LambdaDelta-1/csuba/arity.ma".
-
-include "LambdaDelta-1/arity/lift1.ma".
-
-include "LambdaDelta-1/arity/aprem.ma".
-
-include "LambdaDelta-1/llt/props.ma".
-
-include "LambdaDelta-1/drop1/getl.ma".
-
-include "LambdaDelta-1/drop1/props.ma".
-
-include "LambdaDelta-1/lift1/props.ma".
-
-theorem sc3_arity_gen:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c
-t) \to (arity g c t a)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind
-(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
-t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity
-g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_:
-(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to
-(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity
-g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in
-(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
-a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
-Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity
-g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
-a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
-Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
-
-theorem sc3_repl:
- \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c
-t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c:
-C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3
-g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3:
-A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to
-(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c:
-C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3
-g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall
-(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3
-c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda
-(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c
-t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0
-in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda
-(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
-(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n
-n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k)
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0:
-nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort
-n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1
-x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in
-(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0)
-H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity
-g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2))))))))))
-(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c:
-C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
-(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to
-(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0:
-A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c:
-C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to
-(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t)
-\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1:
-((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t:
-T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c
-t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t
-(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
-(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is
-t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4
-\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
-T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
-(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity
-g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a
-d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat
-Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in
-(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a
-a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4:
-A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0
-x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A
-(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1)
-(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall
-(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d
-c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t
-(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d:
-C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is:
-PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14:
-(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0
-t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0
-(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat
-Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0)
-(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3
-H11))))))) H7))))) H4)))))))))))) a2)) a1)).
-
-theorem sc3_lift:
- \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e
-t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
-\to (sc3 g a c (lift h d t))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e:
-C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t))))))))))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda
-(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in
-(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
-(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n
-n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0))
-(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e
-t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e:
-C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d
-t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t:
-T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e:
-C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall
-(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
-e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c:
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3
-\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall
-(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g
-a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
-(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
-(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
-(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda
-(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
-PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
-t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0:
-C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
-\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t)))))))))
-(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w:
-T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1
-is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1
-(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w
-t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t))
-(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
-
-theorem sc3_lift1:
- \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds:
-PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e)
-\to (sc3 g a c (lift1 hds t)))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds:
-PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g
-a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c:
-C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c
-e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0:
-C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0:
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3
-g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c:
-C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n
-n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x
-in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2
-e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n
-n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0
-H4) c n n0 H3)))) H2))))))))))) hds)))).
-
-theorem sc3_abbr:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i:
-nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads
-(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to
-(sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
-TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
-C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
-(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef
-i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c:
-C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v))
-(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda
-(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g
-c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat
-Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef
-i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2:
-(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda
-(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c
-(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs
-(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2)
-(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda
-(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v:
-T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to
-((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs
-(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs:
-TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c:
-C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c
-(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef
-i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda
-(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs
-(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0
-d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat
-Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda
-(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity
-g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0:
-C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
-\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
-(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead
-a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is:
-PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
-(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads
-(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0:
-C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
-\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
-(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i))
-(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
-(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
-(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs
-(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0
-w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def
-(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C
-(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is
-i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead
-(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x:
-C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i)
-d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w
-(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is
-(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r
-T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w
-(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans
-is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1
-d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T
-(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1
-d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1
-is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v)
-vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v))
-H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs
-(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8)))))))))))
-H3))))))))))))) a)).
-
-theorem sc3_cast:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
-(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t:
-T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl)
-vs (THead (Flat Cast) u t))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
-TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat
-Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to
-(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda
-(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
-T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) |
-(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t:
-T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0))
-(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g
-(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
-(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads
-(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land
-(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0))
-(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1:
-(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2:
-(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat
-Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs
-u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c
-(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads
-(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads
-(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat
-Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs
-t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads
-(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat
-Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat
-Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs
-t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort
-O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))
-(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t
-H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with
-[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c
-(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t)
-(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c
-(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads
-(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1
-n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads
-(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let
-H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))
-(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs
-(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs
-(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u)
-(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def
-H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3
-c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead
-(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead
-(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort
-(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g
-c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c
-(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs
-(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n
-H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall
-(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to
-(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c
-(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1:
-A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3
-g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c
-(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead
-(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
-T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc
-g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
-is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land
-(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall
-(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
-d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3
-\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
-a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
-is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs
-(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3
-g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead
-(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u
-t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0
-(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d
-w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead
-(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2
-in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
-t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
-(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
-(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
-(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity
-g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d:
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
-t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
-(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
-(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
-(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c
-u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9:
-(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y
-\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1
-is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d
-(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1
-is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl)
-(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat
-Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w
-t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u))
-(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat
-Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w
-H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl
-is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t))
-(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl
-is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
-
-theorem sc3_props__sc3_sn3_abst:
- \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g
-a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def
-(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
-((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c:
-C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs:
-TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in
-(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
-(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall
-(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3
-c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
-(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to
-((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n
-n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c:
-C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
-t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
-t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2))
-H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H:
-(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0:
-(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat
-Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H
-(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land
-(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall
-(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl)
-vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c
-(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0:
-(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t))))
-(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
-(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
-(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c:
-C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d:
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t))))
-(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
-(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c
-vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))
-(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads
-(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t:
-T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall
-(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
-d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind
-(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0))))
-(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads
-(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to
-(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_:
-((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0
-t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
-C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i))
-\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef
-i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0:
-T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i:
-nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to
-((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs
-(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0:
-T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs:
-TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs
-(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0
-(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind
-(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
-\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
-(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0
-a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
-(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
-(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0)
-in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d:
-C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d:
-C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t)
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2
-O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10
-(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1)
-(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0
-H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1))
-I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1)
-(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0
-(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil
-(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst)
-x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3
-(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S
-x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef
-O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O
-t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop
-(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2)))))
-(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g
-c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c
-(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl)
-vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
-\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
-(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d:
-C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is:
-PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall
-(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
-TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
-vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0
-c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl)
-w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0:
-C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_:
-((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0
-(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3
-c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9
-\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to
-(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0:
-C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef
-i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef
-i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
-(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t)
-\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0:
-nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1)
-\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat
-Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs)))
-in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i)))
-(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef
-(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat
-Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i))
-(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
-is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i)))
-(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0
-(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1
-(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1))
-(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is
-(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is
-(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2)
-(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is
-vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i))
-(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat
-Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
-
-theorem sc3_sn3:
- \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c
-t) \to (sn3 c t)))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
-(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
-H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
-c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g
-c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0
-vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t)
-(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0
-t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
-C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i))
-\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef
-i))))))))))).(H1 c t H))) H0))))))).
-
-theorem sc3_abst:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
-(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef
-i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda
-(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i))
-a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
-(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0:
-C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
-TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
-vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a
-c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl)
-vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t)
-\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0:
-nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to
-((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl)
-vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))).
-
-theorem sc3_bind:
- \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1:
-A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v:
-T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts
-(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs
-(THead (Bind b) v t)))))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
-(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall
-(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads
-(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads
-(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0:
-nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t:
-T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
-(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat
-Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0
-in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O
-vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S
-O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t))
-(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda
-(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
-(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl)
-(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind
-b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))
-(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0)
-H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2))))))))))
-(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall
-(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl)
-(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl)
-vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall
-(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead
-c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v)
-\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v
-t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda
-(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
-(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a
-d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g
-a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs)
-t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind
-(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
-(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
-(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat
-Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land
-(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0))
-(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
-(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c
-(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda
-(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
-PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl)
-w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity
-g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d:
-C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c)
-\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead
-(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1
-H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3
-g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def
-(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is
-vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead
-(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is)
-t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl)
-(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList
-(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d
-(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat
-Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl)
-(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is
-v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is
-v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S
-O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is)
-(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts
-(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O
-vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is
-d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is
-(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead
-(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
-
-theorem sc3_appl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs:
-TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads
-(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w:
-T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead
-(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a:
-A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3
-g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v)
-\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat
-Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda
-(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v:
-T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs
-(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead
-(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda
-(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads
-(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat
-Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs
-(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads
-(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3:
-(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n
-n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v
-t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead
-(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat
-Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen
-g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3)
-(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1)))))
-H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall
-(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs
-(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g
-(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v
-(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall
-(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c
-(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to
-(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl)
-vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs:
-TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land
-(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0))
-(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
-(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c
-v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1
-in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))
-(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall
-(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
-(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c
-(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead
-a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
-(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w
-t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind
-Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0:
-T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
-(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v
-t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v
-(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0:
-T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
-(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v
-(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g
-c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5)
-(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is:
-PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1
-is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda
-(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat
-Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3
-g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0))))
-(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0:
-T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs)
-(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1
-is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead
-(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads
-(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs
-(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0
-t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead
-(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead
-(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t))
-(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d
-w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t))
-(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is
-v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat
-Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v
-(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr3/defs.ma".
-
-inductive sn3 (c: C): T \to Prop \def
-| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
-
-definition sns3:
- C \to (TList \to Prop)
-\def
- let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil
-\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))])
-in sns3.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sn3/defs.ma".
-
-include "LambdaDelta-1/pr3/props.ma".
-
-theorem sn3_gen_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
-(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
-(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
-T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))
-(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T
-y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0))))
-(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x))
-\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda
-(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
-(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda
-(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
-c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2)
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall
-(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c
-(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T
-t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
-T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1
-x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead
-(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
-(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c
-(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2)
-\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4
-(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
-b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
-(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x
-x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
-T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T
-x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b)
-t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2)
-(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda
-(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b)
-x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
-Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4
-(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
-b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x
-x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
-T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T
-t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in
-(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0
-t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3
-(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c
-x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y
-H0))))) H))))).
-
-theorem sn3_gen_flat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
-(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
-(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0:
-T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y:
-T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
-(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0:
-T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3
-c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T
-t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1:
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land
-(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
-(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
-T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1
-x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in
-(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2)
-\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead
-(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2:
-T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7:
-(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T
-(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let
-H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11
-\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0:
-Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7
-(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T
-(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2)
-(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c
-x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
-Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x
-t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x
-t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0)
-(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
-T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal
-T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f)
-x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c
-x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
-
-theorem sn3_gen_head:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
-(THead k u t)) \to (sn3 c u)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u:
-T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b:
-B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
-(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in
-(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
-c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f:
-F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
-(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in
-(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
-(sn3 c t)).H1)) H0)))))))) k).
-
-theorem sn3_gen_cflat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead
-c (Flat f) u) t) \to (sn3 c t)))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
-(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0:
-T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
-(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T
-t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
-(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2)
-\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2
-(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
-
-theorem sn3_gen_lift:
- \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1
-(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
-\def
- \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1
-t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))
-(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq
-T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0)))))
-(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to
-(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1:
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to
-(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1
-c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d
-x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T
-t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0))
-\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d
-x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq
-T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2)))))
-H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T
-x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d
-t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let
-H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h
-d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to
-(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T
-x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2
-H4)))))))))))))) y H0)))) H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sn3/props.ma".
-
-include "LambdaDelta-1/drop1/fwd.ma".
-
-include "LambdaDelta-1/lift1/fwd.ma".
-
-theorem sns3_lifts1:
- \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to
-(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
-\def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
-(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c
-(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda
-(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H)
-in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList
-ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c
-H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
-(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to
-(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0
-p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def
-(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
-(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1
-(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda
-(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t:
-TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1))
-(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sn3/defs.ma".
-
-include "LambdaDelta-1/nf2/dec.ma".
-
-include "LambdaDelta-1/nf2/pr3.ma".
-
-theorem sn3_nf2:
- \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t
-(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P:
-Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2
-H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y)
-in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P:
-Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3
-(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
-
-theorem nf2_sn3:
- \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c
-t u)) (\lambda (u: T).(nf2 c u)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda
-(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u))))
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let
-H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2
-c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u))
-(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
-T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall
-(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c
-t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1
-x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1
-x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u:
-T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1
-u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x
-x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u))
-(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3))
-H2)))))) t H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sn3/nf2.ma".
-
-include "LambdaDelta-1/sn3/fwd.ma".
-
-include "LambdaDelta-1/nf2/iso.ma".
-
-include "LambdaDelta-1/pr3/iso.ma".
-
-theorem sn3_pr3_trans:
- \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1
-t2) \to (sn3 c t2)))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda
-(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2:
-T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
-(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3:
-T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3:
-(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let
-H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T
-t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let
-H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8
-\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P)))
-H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2
-H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P:
-Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
-
-theorem sn3_pr2_intro:
- \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to
-(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1
-t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c
-t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to
-(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in
-((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall
-(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3
-c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0)))))
-(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall
-(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t
-t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t))))))
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5:
-T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3
-t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to
-((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7:
-((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4
-t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P:
-Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq
-T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10:
-(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to
-(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t:
-T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t
-t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t:
-T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3)
-\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5))
-H9))))))))))) t1 t2 H1 H3)) H2)))))))).
-
-theorem sn3_cast:
- \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to
-(sn3 c (THead (Flat Cast) u t))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda
-(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0)))))
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2
-t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0:
-T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3:
-((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0
-t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1
-t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2:
-T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P:
-Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def
-(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c
-t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0
-x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12
-\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to
-(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T
-(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def
-(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1)
-\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14:
-(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat
-Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1
-H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1
-H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1)))
-(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1)
-((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1))
-(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3:
-T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall
-(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3:
-T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead
-(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c
-(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to
-(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14))))
-(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda
-(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda
-(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17
-\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead
-(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18
-\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16
-(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec
-t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to
-(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def
-(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat
-Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def
-(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0
-(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16:
-(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1
-H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0
-t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8)))
-H7))))))))) t H2)))))) u H))).
-
-theorem sn3_cflat:
- \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u:
-T).(sn3 (CHead c (Flat f) u) t)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f:
-F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1
-(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P:
-Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2
-(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
-
-theorem sn3_shift:
- \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c
-(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
-(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
-H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
-(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
-v) t)).H2)) H0))))))).
-
-theorem sn3_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
-T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3
-(CHead c (Bind b) v2) t)))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
-(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda
-(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind
-b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2)
-\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3
-(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
-(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1
-(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P:
-Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
-(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
-v1)))))))))) t H0))))))).
-
-theorem sn3_gen_def:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
-i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
-(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
-i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
-Abbr c d v i H))))))).
-
-theorem sn3_cdelta:
- \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
-(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
-C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
-\def
- \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
-T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
-\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
-C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
-(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
-(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
-(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
-\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
-C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
-v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
-(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
-(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
-(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
-c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
-H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
-(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
-(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
-v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
-C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
-v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
-(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
-c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
-(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
-C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
-(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
-(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3
-(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
-(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
-(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
-H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
-t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
-(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
-in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda
-(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
-H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
-(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
-C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
-v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
-(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
-(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
-H0)))))).
-
-theorem sn3_cpr3_trans:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
-(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
-t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
-u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1)
-t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0))
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1)
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
-t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T
-t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1
-t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
-
-theorem sn3_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t:
-T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c
-u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t)
-t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_:
-((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
-t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c
-(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t:
-T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b)
-t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2:
-T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b)
-t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b)
-t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda
-(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda
-(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst)
-in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3)
-(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c
-(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b
-(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P:
-Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall
-(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
-b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let
-H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3
-(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
-\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to
-(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def
-(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0
-x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3
-(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P:
-Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind
-Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in
-(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P:
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let
-H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2)
-(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let
-H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in
-(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1
-\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda
-(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T
-(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P:
-Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0:
-T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0))))
-H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1
-t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst)
-t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P:
-Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20:
-(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1)
-in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P:
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let
-H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0:
-T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda
-(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans
-c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1
-H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20
-H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst
-t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b
-Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0
-in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind
-b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
-(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b)
-t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq
-T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13:
-(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0:
-T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead
-(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
-T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in
-(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead
-(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0
-(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to
-(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda
-(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c
-(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def
-H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c
-(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r
-T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0))
-\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1
-(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T
-t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead
-(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20:
-(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0
-H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2
-\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18:
-(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c
-(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c
-(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind
-b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq
-T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1
-x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
-(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
-t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
-(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10)
-c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t
-H2)))))) u H)))).
-
-theorem sn3_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
-t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
-(Bind Abst) w t))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
-(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
-c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
-Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
-y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst)
-w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead
-(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
-Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall
-(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w:
-T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w
-x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2)
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda
-(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
-c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x
-x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead
-(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
-(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c
-w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1:
-T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0:
-T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0
-x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1
-(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P)))
-\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in
-(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0
-x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to
-(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8:
-((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2
-t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3
-x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
-(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind
-Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead
-(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x
-(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
-(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
-(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst)
-t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq
-T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15:
-(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda
-(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to
-(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T
-(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def
-(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2))
-(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst)
-x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind
-T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
-(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
-x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
-(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
-H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c
-(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2
-x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl)
-x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0
-x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3
-(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0:
-T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def
-(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
-(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2
-x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl)
-t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let
-H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind
-T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2
-x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T
-x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
-(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def
-(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind
-Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall
-(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0
-H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead
-(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind
-Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
-H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
-(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind
-Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in
-(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0)
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
-(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27:
-(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4)
-(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x
-x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
-H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
-x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2
-H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P:
-Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind
-(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl)
-x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x
-(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))))
-(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4)
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead
-(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T
-x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
-Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4
-H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
-(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind
-Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in
-(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0)
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
-(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25)))
-(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind
-Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr)
-x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x
-x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
-H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
-x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23
-(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13)))))))
-H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead
-(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3
-x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0)
-\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T
-(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0]))
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0)
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in
-(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0
-H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x
-x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4))
-(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0:
-T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead
-(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in
-(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead
-(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0
-t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr)
-x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26:
-(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4)
-(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x
-x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0:
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
-(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2
-c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4
-(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3)
-\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq
-T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P:
-Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
-(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr)
-x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
-(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda
-(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29
-(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15)
-(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3)))))
-H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
-(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14:
-(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq
-T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
-(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c
-(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
-Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
-H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
-x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
-(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _)
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in
-((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14)
-in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0
-H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2
-H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b)
-x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b:
-B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3
-c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29
-\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_:
-False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5)
-x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12))
-H11))))))))) w H4))))))))))) y H0))))) H)))).
-
-theorem sn3_appl_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v:
-T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
-(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead
-(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef
-i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2:
-T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall
-(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i))
-t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
-(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1
-x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda
-(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P)))
-H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1)
-(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq
-T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
-Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda
-(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i)
-(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1
-x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall
-(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T
-t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat
-Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P:
-Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c
-t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t
-(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c
-(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0)
-\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H
-x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))
-(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8:
-(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_:
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let
-H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i))
-t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r
-T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind
-T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c
-(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B
-T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
-Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1:
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead
-(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat
-Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2
-c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def
-(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to
-(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
-O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
-O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0)
-x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6))
-H5))))))))) v H0))))).
-
-theorem sn3_appl_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v
-(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
-(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
-(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead
-(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro
-T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3
-c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x:
-T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat
-Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2:
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
-t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
-Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
-i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift
-(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
-T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall
-(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead
-(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w))
-H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2
-(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat
-Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl)
-x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead
-(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8)
-in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
-(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c
-x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P:
-Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat
-Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i
-H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u:
-T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
-T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16:
-(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead
-(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
-Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c
-(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x
-in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
-(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def
-(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead
-(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T
-(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0
-H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead
-(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x
-(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P:
-Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
-(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead
-(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0
-(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let
-H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22
-(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w))
-(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl)
-(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O
-w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr)
-x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0
-t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T
-(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20
-\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H
-(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2
-(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3)
-(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in
-((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d
-(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w)
-i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24
-\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20
-w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S
-i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0
-(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def
-H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
-(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c
-(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x
-x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w))
-(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat
-Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t]))
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
-w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to
-(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda
-(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
-w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3
-H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10:
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
-t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
-b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0
-x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c
-x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
-u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
-Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2
-x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t))
-(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
-x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2
-H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i)
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
-(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
-Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda
-(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
-x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t:
-T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in
-(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
-(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead
-(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10))
-H9))))))))))))) y H1)))) H0))))))).
-
-theorem sn3_appl_cast:
- \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v
-u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead
-(Flat Appl) v (THead (Flat Cast) u t))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead
-(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3
-c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
-(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y:
-T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead
-(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
-(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v
-(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to
-(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat
-Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall
-(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0:
-T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead
-(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2:
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2
-(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x
-t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
-t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
-(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x
-t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0))
-(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t))))
-(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq
-T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat
-Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0
-(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
-x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2)
-\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda
-(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3
-c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c
-(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1:
-T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T
-t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat
-Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
-x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0
-(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
-\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let
-H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to
-(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3:
-T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead
-(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3
-t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1
-(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
-\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
-(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda
-(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0
-x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat
-Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x
-(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
-(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1)
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T
-T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
-(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast)
-x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq
-T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19:
-(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda
-(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to
-(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T
-(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def
-(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c
-x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2
-x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat
-Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1
-x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
-(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
-Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat
-Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x
-\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let
-H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
-x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall
-(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5)))
-(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
-x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
-(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl)
-x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
-\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0)
-(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def
-(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat
-Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall
-(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3:
-T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead
-(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2
-(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))
-(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P)))
-H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18
-x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
-(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1)
-(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat
-Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead
-(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
-(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1)
-(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
-\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1)
-(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3:
-T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl)
-x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let
-H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in
-(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast)
-x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))
-(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda
-(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall
-(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat
-Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5
-(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29)))
-(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4))
-\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x
-x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead
-(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1)
-(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat
-Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl)
-x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
-(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl)
-x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1
-| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
-Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x
-x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32)
-in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat
-Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
-(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28
-x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x
-H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
-(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead
-(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl))
-x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat
-Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead
-(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P:
-Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x
-x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat
-Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2
-c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23)))))))
-H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat
-Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T
-(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl)
-x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead
-(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead
-(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
-\Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1)
-(THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 |
-(TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
-Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x
-x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26)
-in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
-(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
-Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat
-Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
-(Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to
-(\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda
-(t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c
-(THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2
-H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1)
-(THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat
-Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3
-H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
-u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2)
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
-(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda
-(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x
-x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
-u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead
-(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13
-(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5)
-(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0
-x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2
-x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2
-H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat
-Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2)
-(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
-(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18:
-(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq
-T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda
-(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c
-(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T
-(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P:
-Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5))
-H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6)
-x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast)
-x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4)
-H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O)
-O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5))))
-H4))))))))) y H0))))) H)))).
-
-theorem sn3_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
-T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u)
-(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v
-(THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
-(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0:
-T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O)
-O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0)))))))
-(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat
-Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2
-t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
-(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
-t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))))
-(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t
-(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c
-(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0:
-T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3
-c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b)
-t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat
-Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b)
-t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3)
-\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3
-(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2
-t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to
-(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
-x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1
-x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead
-(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0:
-T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3
-(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
-(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1
-(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0)
-H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
-t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
-(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0)
-H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda
-(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0))
-t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x
-(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b)
-t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
-t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0)
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3)
-(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0)
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c
-t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat
-Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead
-(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
-H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2)
-(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in
-(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind
-b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda
-(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
-(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
-(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3
-x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1)
-x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl)
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P:
-Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3
-x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def
-(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3)
-\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3
-x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0:
-T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1
-(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T
-t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4))))
-(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4)
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead
-(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead
-(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0
-H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b)
-t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
-Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32
-\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3
-c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x
-x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl)
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
-(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda
-(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c
-(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead
-(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead
-(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P:
-Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O
-H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
-(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0
-(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29))))
-(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat
-Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S
-O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P:
-Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
-\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in
-(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
-H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead
-(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b)
-t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r
-T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let
-H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead
-(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall
-(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O
-H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b)
-t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S
-O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15))
-x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1)
-x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P:
-Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead
-(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26
-\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P))
-(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda
-(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead
-c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2
-\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x
-x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl)
-(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T
-x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0:
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
-x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O
-H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
-(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
-H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P:
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in
-(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
-H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c
-(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda
-(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal
-T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift
-(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c
-c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26))))))
-H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift
-(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2)
-(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat
-Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def
-(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
-O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1
-(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0:
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
-x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O
-H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
-(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
-H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx
-(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift
-(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c
-(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13))
-(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0:
-T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
-T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4)))))))
-(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1
-x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
-x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind
-b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
-(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10
-(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4)
-(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in
-((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14)
-in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def
-(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead
-c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda
-(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind
-Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def
-(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl)
-(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
-b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind
-b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0:
-B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl)
-(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4
-(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5
-(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
-\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind
-b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat
-Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def
-(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30
-\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_:
-False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20))
-H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b)
-t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
-Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1:
-B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T
-(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3
-(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
-(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead
-c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
-H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in
-(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
-(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
-((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
-(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0
-H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1
-H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0)
-x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind
-b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead
-(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1
-(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def
-(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
-O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5
-(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0:
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let
-H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq
-T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat
-Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
-H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat
-Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat
-Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to
-(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda
-(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
-(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in
-(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0:
-Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2
-(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P))))))
-(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O
-x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c
-(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
-(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x)
-Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5)
-x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in
-(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def
-(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
-H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda
-(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
-H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2
-c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
-(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x)
-(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind
-b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c
-(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat
-Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl)
-(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O
-x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O
-x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13))
-H12)))))))))))))) y H4))))) H3))))))) u H0))))).
-
-theorem sn3_appl_appl:
- \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
-(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
-(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
-(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
-u1)))))))))
-\def
- \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
-(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T
-(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall
-(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2)
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
-(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
-y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to
-(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso
-y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2))))))
-\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t:
-T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2:
-T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
-(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x:
-T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2:
-T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
-(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
-(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
-(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T
-t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
-(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
-(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
-t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2
-(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c
-v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso
-t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2))))))
-\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5:
-((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0
-t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to
-(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2
-u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
-Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7:
-((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P:
-Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T
-t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead
-(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall
-(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to
-(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P:
-Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat
-Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T
-t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P:
-Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
-(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall
-(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
-(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3
-t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2
-(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P)))
-\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
-(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat
-Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x
-x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead
-(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c
-(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def
-(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl)
-x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
-(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl)
-x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T
-t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18:
-(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t:
-T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
-Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat
-Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2
-H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c
-(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1
-x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat
-Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0
-x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0
-(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
-Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat
-Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
-(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26
-\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))
-((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P:
-Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda
-(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
-\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in
-((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27)
-in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t:
-T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
-x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let
-H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind
-T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))))
-(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
-(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))
-\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3
-(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c
-(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0
-x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
-(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0)))
-(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t:
-T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
-t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let
-H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind
-T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))
-(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c
-(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36:
-(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17)
-(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda
-(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead
-(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1
-u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4
-H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat
-Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27
-(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4
-(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c
-t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c
-(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3
-x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0
-u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0)
-(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4)
-(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28))
-(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29
-(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x
-x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead
-(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat
-Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat
-Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
-(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23:
-(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda
-(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4
-x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl)
-t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
-Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind
-Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def
-(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl)
-x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P:
-Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
-t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
-(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall
-(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
-(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4)
-H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c
-(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead
-(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t:
-T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c
-t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso
-(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
-(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
-t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat
-Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing
-c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4))
-(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind
-Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind
-Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6
-(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5
-x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t:
-T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
-(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to
-P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1)
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort
-n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda
-(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T
-(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2)
-\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind
-Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5
-x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
-(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind
-Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow
-(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst)
-x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5
-x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
-| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
-(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 |
-(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4)
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3
-x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T
-t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr)
-x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_:
-T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5)
-(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind
-Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T
-(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40:
-(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def
-(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq
-T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k
-(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat
-Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5
-x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead
-(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind
-Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5
-x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
-(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda
-(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22:
-(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4
-x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S
-O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4
-x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind
-T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
-(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8
-(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c
-(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T
-(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead
-(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P:
-Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
-t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
-(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall
-(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
-(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23)
-in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead
-(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
-Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4
-x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4:
-T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to
-(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
-(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
-Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
-t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat
-Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32
-(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c
-(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat
-Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead
-(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x)
-x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl
-x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
-(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift
-(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c
-(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3)
-x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x
-x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl)
-(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl
-(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind
-x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
-x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t:
-T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
-(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat
-Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow
-(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4
-x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl)
-(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T
-(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
-P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef
-i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T
-(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
-x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
-(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) |
-(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4)
-(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k
-v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let
-H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t)
-\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4
-x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4
-| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
-(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat
-Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0:
-K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0
-v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
-P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4
-(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq
-T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_:
-T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl)
-(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5
-t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43
-\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
-H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4
-(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))])
-in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5)))
-(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
-x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift
-(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead
-(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8
-(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat
-Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))))))))))
-x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1:
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T
-(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3
-(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_:
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let
-H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead
-(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3
-x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t))
-(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind
-Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
-(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17:
-(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T
-t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
-(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c
-(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T
-(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
-Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
-H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
-x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x
-x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3)
-H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O)
-O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2
-H4))))))))) y H0))))) H))))).
-
-theorem sn3_appl_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
-(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
-\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w
-t))))))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
-(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
-T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
-Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind
-Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
-Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
-Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
-H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead
-(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind
-Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
-Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c
-(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl))))))))
-H1))))))))).
-
-theorem sn3_appl_appls:
- \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads
-(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall
-(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2)
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
-(sn3 c (THead (Flat Appl) v2 u1))))))))))
-\def
- \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads
-(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead
-(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0:
-(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1
-(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads
-(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
-Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0
-H1))))))))).
-
-theorem sn3_appls_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us:
-TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
-(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads
-(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H))
-(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3
-c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t)
-(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef
-i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil
-(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1
-in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
-TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref
-c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
-(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land
-(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2
-(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads
-(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3
-c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1)
-(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
-(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c
-t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
-(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda
-(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1)
-(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat
-Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl)
-(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9
-(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t
-u2))))))))) H5))) H3))))))) t0))) us)))).
-
-theorem sn3_appls_cast:
- \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat
-Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3
-c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
-\def
- \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
-(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads
-(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u
-t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda
-(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0:
-TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads
-(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2))
-\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to
-(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to
-(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2)))
-\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u
-t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil
-u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c
-(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u:
-T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil
-u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
-(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1:
-T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat
-Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to
-(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall
-(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall
-(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c
-(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u
-t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl)
-(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1
-t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u
-t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
-(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead
-(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def
-(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3
-\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat
-Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
-(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c
-(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let
-H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in
-(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads
-(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1
-t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c
-(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in
-(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8
-(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2)
-(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl)
-(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P:
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons
-t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl)
-(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t
-Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
-
-theorem sn3_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
-T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind
-b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat
-Appl) vs (THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
-(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t:
-TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts
-(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0))))))
-(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u
-H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t:
-TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
-(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u
-t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl)
-(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c
-(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0))))))))
-(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
-(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b)
-u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil)
-t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0:
-TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads
-(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead
-(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to
-(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u
-t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u)
-(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads
-(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1:
-T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
-v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
-(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
-(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3
-(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat
-Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c
-(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
-t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda
-(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads
-(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c
-(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c
-(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2:
-T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
-t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind
-b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b
-H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v
-(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))
-(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0))
-t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead
-(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9
-Appl)))))))))) H4))))))))) vs0))) vs)))))).
-
-theorem sn3_appls_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
-(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c
-w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst)
-w t))))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us:
-TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead
-(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H:
-(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c
-w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0:
-TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0
-(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads
-(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3
-c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_:
-(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w:
-T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead
-(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads
-(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1:
-(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1:
-TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v
-t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead
-(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u
-(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c
-w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl)
-v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat
-Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
-\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
-(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
-T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
-(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
-(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl)
-(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4:
-(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
-(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead
-(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3
-c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w
-t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat
-Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8
-\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
-(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
-t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
-t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
-
-theorem sn3_lift:
- \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h:
-nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
-\def
- \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda
-(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d)
-\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d
-t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall
-(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c:
-C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c
-d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T
-(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i
-t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T
-(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3))
-(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda
-(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h
-i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T
-(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1
-x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T
-(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let
-H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10
-(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6)))))
-H5))))))))))))) t H))).
-
-theorem sn3_abbr:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d
-v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef
-i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let
-H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T
-(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u))))
-(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2)
-(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t:
-T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in
-(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i))
-(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0:
-C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda
-(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr)
-x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2
-(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S
-i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let
-H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H
-(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0
-(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1)
-(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in
-((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d
-(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v)
-i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12
-\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v
-H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def
-(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d
-H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10))))
-H9))) t2 H6)))))) H4)) H3))))))))))).
-
-theorem sn3_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl)
-vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
-(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3
-c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O
-w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H))
-in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0:
-TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift
-(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c
-(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c
-(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_:
-(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat
-Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads
-(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1)))
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat
-Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i)))))
-\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w))))
-\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef
-i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i)
-O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
-(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
-O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
-t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead
-(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4:
-(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0
-(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda
-(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i))
-u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to
-(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat
-Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2)
-(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2
-(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl))))))))
-H3)))))))) vs0))) vs)))))).
-
-theorem sns3_lifts:
- \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h
-i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda
-(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t:
-TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0)
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c
-(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def
-H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
-(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj
-(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
-H4)))) H2)))))) ts)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/theory.ma".
-
-axiom pc3_gen_appls_sort_abst:
- \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall
-(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u))
-\to False)))))
-.
-
-axiom pc3_gen_appls_lref_abst:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall
-(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to
-False))))))))
-.
-
-axiom pc3_gen_appls_lref_sort:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws:
-TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads
-(Flat Appl) ws (TSort n))) \to False))))))))
-.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/G/defs.ma".
-
-include "LambdaDelta-1/getl/defs.ma".
-
-inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def
-| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort
-(next g n))))
-| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w)
-\to (sty0 g c (TLRef i) (lift (S i) O w))))))))
-| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w)
-\to (sty0 g c (TLRef i) (lift (S i) O v))))))))
-| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
-T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead
-(Bind b) v t1) (THead (Bind b) v t2)))))))
-| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2:
-T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat
-Appl) v t2))))))
-| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1
-v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c
-(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sty0/defs.ma".
-
-theorem sty0_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c
-(TSort n) x) \to (eq T x (TSort (next g n)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
-(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c
-t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda
-(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda
-(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_:
-C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1:
-nat).(eq T (TSort (next g n1)) (TSort (next g n)))) (refl_equal T (TSort
-(next g n))) n0 H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v:
-T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
-v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v
-(TSort n)) \to (eq T w (TSort (next g n)))))).(\lambda (H4: (eq T (TLRef i)
-(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
-(False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl
-i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
-w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g
-n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T
-(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
-\Rightarrow False])) I (TSort n) H4) in (False_ind (eq T (lift (S i) O v)
-(TSort (next g n))) H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda
-(v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind
-b) v) t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g
-n)))))).(\lambda (H3: (eq T (THead (Bind b) v t1) (TSort n))).(let H4 \def
-(eq_ind T (THead (Bind b) v t1) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
-(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4))))))))))
-(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort
-(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let
-H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
-(False_ind (eq T (THead (Flat Appl) v t2) (TSort (next g n))) H4)))))))))
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
-v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 (TSort (next g
-n)))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g
-n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort n))).(let H6
-\def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
-(False_ind (eq T (THead (Flat Cast) v2 t2) (TSort (next g n))) H6))))))))))))
-c y x H0))) H))))).
-
-theorem sty0_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c
-(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T x (lift (S n) O u)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
-(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c
-t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u:
-T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u
-t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O
-u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda
-(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C
-T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u
-t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n)
-O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
-n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda
-(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_:
-T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0:
-nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort
-n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (e:
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (TSort (next g n0)) (lift (S n)
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
-n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
-T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T
-(TSort (next g n0)) (lift (S n) O u))))))) H2))))) (\lambda (c0: C).(\lambda
-(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d
-(Bind Abbr) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_:
-(((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u:
-T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T w (lift (S n) O
-u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 \def (f_equal T
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i]))
-(TLRef i) (TLRef n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0:
-nat).(getl n0 c0 (CHead d (Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n
-(\lambda (n0: nat).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (lift (S n0) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n0) O w) (lift (S n) O
-u)))))))) (or_introl (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (lift (S n) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e:
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O w) (lift (S n) O
-u)))))) (ex3_3_intro C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (lift (S n) O w) (lift (S n) O t))))) d v w H6 H2 (refl_equal T
-(lift (S n) O w)))) i H5)))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind
-Abst) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T
-v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T w (lift (S n) O u)))))))))).(\lambda (H4: (eq T
-(TLRef i) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in
-T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0)
-\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H4) in
-(let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind
-Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C T T
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v)
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v)
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v)
-(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i
-H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1
-t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e:
-C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e
-u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n)
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
-n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T
-(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v
-t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Bind b) v t2) (lift (S
-n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(eq T (THead (Bind b) v t2) (lift (S n) O u))))))) H4))))))))))
-(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O
-t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
-T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2
-(lift (S n) O u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef
-n))).(let H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
-H3) in (False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O t)))))) (ex3_3 C T T
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Flat
-Appl) v t2) (lift (S n) O u))))))) H4))))))))) (\lambda (c0: C).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T v2 (lift (S n) O u)))))))))).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H5: (eq T
-(THead (Flat Cast) v1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat
-Cast) v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat
-Cast) v2 t2) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u)))))))
-H6)))))))))))) c y x H0))) H))))).
-
-theorem sty0_gen_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
-T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda
-(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead
-(Bind b) u t2))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
-T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1)
-x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x))
-(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2))
-(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda
-(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
-(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g
-(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u
-t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
-(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind b) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g
-(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n))
-(THead (Bind b) u t2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
-v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v
-(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind b)
-u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda (H4:
-(eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2:
-T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O
-w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T
-v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind
-b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda
-(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2:
-T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O
-v) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (b0: B).(\lambda (c0:
-C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g
-(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind
-b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda
-(H3: (eq T (THead (Bind b0) v t0) (THead (Bind b) u t1))).(let H4 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t0) (THead
-(Bind b) u t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _)
-\Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t0) (THead
-(Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead
-(Bind b) u t1) H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0
-b)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1))
-\to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u)
-t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in
-(let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t
-t2)) H1 t1 H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead
-(Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind
-b0) t) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u
-t3)))))) H9 u H7) in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead
-c0 (Bind b0) t) t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T
-(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T
-(THead (Bind b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0
-(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3:
-T).(sty0 g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3:
-T).(eq T t2 (THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B
-b0 (\lambda (b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in
-(eq_ind_r B b (\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0
-(Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead
-(Bind b) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
-u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u
-t3))) t2 H14 (refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5))
-H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3))
-(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T
-(THead (Flat Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T
-(THead (Flat Appl) v t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u t1) H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
-u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u
-t3)))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u
-t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2))
-(\lambda (t2: T).(eq T v2 (THead (Bind b) u t2))))))).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0
-(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b)
-u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda
-(H5: (eq T (THead (Flat Cast) v1 t0) (THead (Bind b) u t1))).(let H6 \def
-(eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex2 T (\lambda (t3:
-T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat
-Cast) v2 t2) (THead (Bind b) u t3)))) H6)))))))))))) c y x H0))) H))))))).
-
-theorem sty0_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x:
-T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g
-c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x:
-T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead
-(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T
-(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat
-Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
-u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n:
-nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H1) in
-(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-(TSort (next g n)) (THead (Flat Appl) u t2)))) H2))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
-w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
-T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u
-t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5
-\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in
-(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-(lift (S i) O w) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0
-(CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
-w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2:
-T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u
-t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5
-\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in
-(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-(lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (b:
-B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0
-(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind
-b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u
-t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u
-t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T (\lambda (t3:
-T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v t2) (THead
-(Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda
-(t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 t2)).(\lambda (H2: (((eq
-T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3))
-(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))))))).(\lambda (H3: (eq T
-(THead (Flat Appl) v t0) (THead (Flat Appl) u t1))).(let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t]))
-(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in
-(\lambda (H6: (eq T v u)).(let H7 \def (eq_ind T t0 (\lambda (t: T).((eq T t
-(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3))
-(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))))) H2 t1 H5) in (let H8
-\def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T
-u (\lambda (t: T).(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3:
-T).(eq T (THead (Flat Appl) t t2) (THead (Flat Appl) u t3))))) (ex_intro2 T
-(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl)
-u t2) (THead (Flat Appl) u t3))) t2 H8 (refl_equal T (THead (Flat Appl) u
-t2))) v H6))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2:
-T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl)
-u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T
-v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to
-(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead
-(Flat Appl) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t1)
-H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3:
-T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Appl) u t3)))) H6))))))))))))
-c y x H0))) H)))))).
-
-theorem sty0_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall
-(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2:
-T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0
-g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2
-t2))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda
-(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T
-(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_:
-T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda
-(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2:
-T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0
-g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq
-T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_:
-T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2)))
-(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2)))))))))
-(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat
-Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in
-T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast)
-v1 t1) H1) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g
-c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda
-(v2: T).(\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Cast) v2
-t2))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w:
-T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1
-t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g d v1 v2)))
-(\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda
-(t2: T).(eq T w (THead (Flat Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef
-i) (THead (Flat Cast) v1 t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda (v2:
-T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0
-g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O w) (THead
-(Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
-Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T
-v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_:
-T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2)))
-(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2
-t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5
-\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in
-(False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2)))
-(\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2:
-T).(\lambda (t2: T).(eq T (lift (S i) O v) (THead (Flat Cast) v2 t2)))))
-H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
-(\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind b) v) v1 v2)))
-(\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) t1 t3)))
-(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2
-t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1
-t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T T (\lambda
-(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3:
-T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind
-b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda (c0: C).(\lambda
-(v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T
-(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda
-(t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead
-(Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v t0) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1)
-H3) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1
-v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2:
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Flat Cast) v2
-t3))))) H4))))))))) (\lambda (c0: C).(\lambda (v0: T).(\lambda (v2:
-T).(\lambda (H1: (sty0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat
-Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1
-v3))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v3:
-T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) v3 t2)))))))).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0
-(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
-T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3)))
-(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3
-t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) v0 t0) (THead (Flat Cast)
-v1 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0
-| (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast)
-v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
-| (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast)
-v1 t1) H5) in (\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda
-(t: T).((eq T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3:
-T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0
-g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast)
-v3 t3))))))) H4 t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g
-c0 t t2)) H3 t1 H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t
-(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_:
-T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3)))
-(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2
-v1 H8) in (let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1
-H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3)))
-(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3:
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3
-t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2)))))))))
-H6)))))))))))) c y x H0))) H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sty0/defs.ma".
-
-include "LambdaDelta-1/getl/drop.ma".
-
-theorem sty0_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
-e) \to (sty0 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c)
-\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda
-(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_:
-(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d
-(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g
-c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort
-(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v
-w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0:
-nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0
-w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3:
-(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0
-(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le
-i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0)
-in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0)))
-(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_:
-C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h
-d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1:
-C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0
-x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind
-nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
-(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
-H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
-Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S
-i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w)))
-(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
-d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T
-(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind
-nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i)
-(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S
-i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_:
-nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w))))
-(sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x
-(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i))
-w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
-(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
-i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
-(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
-H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
-h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat
-(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
-O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0
-(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
-c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h)
-(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
-h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
-h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
-i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
-i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
-H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
-(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
-T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h:
-nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift
-h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
-(H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h
-d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def
-(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d
-(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
-O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
-(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g
-c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0:
-C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h
-(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let
-H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
-(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0
-h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0
-(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h
-(minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S
-i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
-h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x
-d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i)
-O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g
-c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h
-(minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0
-(\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i))
-v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead
-x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S
-i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
-(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
-i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
-(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
-H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
-h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat
-(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
-O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0
-(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
-c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h)
-(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
-h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
-h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
-i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
-i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
-H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3
-t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h
-d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s
-(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v
-t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4))
-(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b)
-d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d)
-t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2
-b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift
-h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda
-(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g
-c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
-t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
-(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat
-Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v
-t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d)
-t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s
-(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl)
-d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h
-d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d
-(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d))))))))))))
-(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1
-v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d
-v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3
-t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
-t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4:
-(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s
-(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast)
-v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast)
-d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h
-(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0
-h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
-(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat
-Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast)
-v1 t3 h d))))))))))))))) e t1 t2 H))))).
-
-theorem sty0_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c
-t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2:
-T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda
-(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2))
-(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0:
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
-w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2
-in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g
-c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w
-x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i)
-O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0)))))
-H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
-T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g
-d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex
-T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda
-(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v)
-t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v
-i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v:
-T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b)
-v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v)
-t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0
-(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3)
-t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3
-x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead
-(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0:
-C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0
-t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2
-\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
-T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3:
-(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl)
-v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2))))))))
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
-v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2:
-T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T
-(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda
-(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat
-Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6
-\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
-T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7:
-(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast)
-v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7))))
-H6)))) H4))))))))))) c t1 t H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sty1/props.ma".
-
-include "LambdaDelta-1/cnt/props.ma".
-
-theorem sty1_cnt:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c
-t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt
-t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
-T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3))))))
-(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0
-(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0
-(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n)))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0
-g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda
-(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d
-v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef
-i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v
-x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i)
-t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x
-H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T
-(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def
-H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2))
-(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2)))
-(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt
-x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2:
-T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v)
-(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1))
-(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i
-H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0:
-C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g
-(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g
-(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
-(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda
-(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2)
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead
-c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4:
-T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead
-(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v)))))
-H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3:
-T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4:
-T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in
-(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4))
-(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda
-(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda
-(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v
-t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v
-t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0:
-C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1
-v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2:
-T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2
-t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4:
-T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2
-t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead
-(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda
-(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0
-t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g
-c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat
-Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2)
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1
-x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0
-x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4))
-(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat
-Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sty0/defs.ma".
-
-inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def
-| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2))
-| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c
-t t2) \to (sty1 g c t1 t2)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/sty1/defs.ma".
-
-include "LambdaDelta-1/sty0/props.ma".
-
-theorem sty1_trans:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c
-t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g
-c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g
-c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g
-c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0
-g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
-
-theorem sty1_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
-T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead
-(Bind b) v t1) (THead (Bind b) v t2))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1
-t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead
-(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g
-(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead
-(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_:
-(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v
-t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c
-(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t)
-H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))).
-
-theorem sty1_appl:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
-(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat
-Appl) v t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1
-g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3:
-T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1)
-(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t:
-T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v
-t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t
-t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1
-(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))).
-
-theorem sty1_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
-e) \to (sty1 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c:
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h
-d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1
-t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop
-h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0
-c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1:
-((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to
-(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2:
-(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
-(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3)
-(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
-
-theorem sty1_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c
-t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2:
-T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1
-t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1
-t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2:
-T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))).
-
-theorem sty1_abbr:
- \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w)
-\to (sty1 g c (TLRef i) (lift (S i) O w)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w:
-T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c
-(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v
-t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2
-H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c
-(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t
-t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2)
-(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w
-H0)))))))).
-
-theorem sty1_cast2:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c
-t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T
-(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) v3 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1:
-T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c
-v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat
-Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda
-(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T
-(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1)
-(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast
-g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1
-t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to
-(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead
-(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda
-(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g
-c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T
-(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1
-v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast)
-v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1
-g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def
-(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4:
-T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3:
-T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda
-(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g
-c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat
-Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat
-Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast
-g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift/defs.ma".
-
-definition subst:
- nat \to (T \to (T \to T))
-\def
- let rec subst (d: nat) (v: T) (t: T) on t: T \def (match t with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true
-\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k
-u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))]) in subst.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst/defs.ma".
-
-theorem subst_sort:
- \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort
-k)) (TSort k))))
-\def
- \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort
-k)))).
-
-theorem subst_lref_lt:
- \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T
-(subst d v (TLRef i)) (TLRef i)))))
-\def
- \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i
-d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true
-\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i)))
-(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))).
-
-theorem subst_lref_eq:
- \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v)))
-\def
- \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq
-T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with
-[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift
-i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))).
-
-theorem subst_lref_gt:
- \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T
-(subst d v (TLRef i)) (TLRef (pred i))))))
-\def
- \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d
-i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true
-\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef
-(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred
-i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d)
-(le_bge d i (lt_le_weak d i H)))))).
-
-theorem subst_head:
- \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d:
-nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w
-t)))))))
-\def
- \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d:
-nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst/fwd.ma".
-
-include "LambdaDelta-1/subst0/defs.ma".
-
-include "LambdaDelta-1/lift/props.ma".
-
-theorem subst_lift_SO:
- \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S
-O) d t)) t)))
-\def
- \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq
-T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d:
-nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n)))
-(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T
-(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n))
-(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq
-T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n
-d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n)))
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T
-(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d
-(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T
-(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n)))
-(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0))
-(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T
-t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0)
-(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
-(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef
-(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O)
-(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d
-(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0:
-T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0))
-t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v
-(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O)
-d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k
-t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v
-(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1)))
-(f_equal3 K T T T THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v
-(lift (S O) (s k d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))) (subst d v
-(THead k (lift (S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S
-O) d t0) (lift (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1))
-(lift_head k t0 t1 (S O) d)))))))) t)).
-
-theorem subst_subst0:
- \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0
-d v t1 t2) \to (eq T (subst d v t1) (subst d v t2))))))
-\def
- \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda
-(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3))))))
-(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t:
-T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i)
-(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T
-(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0
-t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t))
-(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0)))
-(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0
-i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i)))
-(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2:
-T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1
-u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t:
-T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t))
-(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k
-(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst
-i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0:
-T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s
-k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t)))
-(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i))
-(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k:
-K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i:
-nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k
-i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst
-i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u
-t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda
-(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T
-(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead
-k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0
-u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u
-t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u
-t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
-(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0
-u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i)
-v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s
-k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r
-T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T
-(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0
-u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i
-v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda
-(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k
-i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4)))
-(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4))
-(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1
-t3 i))))))))))))) d v t1 t2 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/defs.ma".
-
-include "LambdaDelta-1/lift/props.ma".
-
-theorem dnf_dec2:
- \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S
-O) d v))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda
-(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d:
-nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n)
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d
-v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort
-n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T
-(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n:
-nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
-(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T
-(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v:
-T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n)
-(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d
-(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind
-nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0
-w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift
-(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w
-(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S
-O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n)
-(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w)
-(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n
-(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d
-H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
-(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n)
-(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred
-n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
-(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
-t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
-v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda
-(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d)
-in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0
-d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
-v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1)
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O)
-d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0
-(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
-(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S
-O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))))
-(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift
-(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d
-v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w
-t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
-T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w)
-in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift
-(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S
-O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s
-k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda
-(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w
-(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d
-w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0
-t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2))
-(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6)
-(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5))))))
-(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
-(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T
-(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex
-T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O)
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x))
-(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def
-H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T
-(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d
-v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d)
-x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O)
-(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d)
-x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1
-H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
-v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex
-T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x:
-T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in
-(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s
-k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S
-O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
-k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1)
-(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v:
-T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O)
-d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
-(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2
-t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T
-(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v))))
-(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T
-(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda
-(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda
-(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d)
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1)
-(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift
-(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1)
-t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift
-(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3))
-(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
-(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T
-(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x)
-(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
-k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
-T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x)
-(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
-(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v))))
-(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k
-d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d
-x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x)
-(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift
-(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O)
-d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
-
-theorem dnf_dec:
- \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or
-(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
-\def
- \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t
-d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v:
-T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S
-O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t
-(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v:
-T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1
-\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T
-(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d
-v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d
-x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t
-(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t
-(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t
-(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex
-T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d
-v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T
-(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0
-(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v:
-T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x)
-(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d
-x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d
-x)))) t H1))) H0)) H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/lift/defs.ma".
-
-inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
-| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift
-(S i) O v)))
-| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i:
-nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v
-(THead k u1 t) (THead k u2 t))))))))
-| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1:
-T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i
-v (THead k u t1) (THead k u t2))))))))
-| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i:
-nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2:
-T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2
-t2)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/defs.ma".
-
-include "LambdaDelta-1/lift/props.ma".
-
-theorem subst0_gen_sort:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
-i v (TSort n) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n)
-(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda
-(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda
-(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_:
-T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def
-(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2)))))
-(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n))
-\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t)
-(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
-(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2:
-T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda
-(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0:
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0
-i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k:
-K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k
-u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))).
-
-theorem subst0_gen_lref:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
-i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0
-i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v))))
-(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0:
-nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n))
-\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0:
-T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0:
-nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj
-(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n)
-(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2:
-T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
-u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
-(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T
-(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2
-t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0:
-T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0
-(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s
-k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T
-(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u
-t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
-u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
-(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef
-n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda
-(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2
-t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
-
-theorem subst0_gen_head:
- \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
-(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
-u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
-T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1)
-x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda
-(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
-T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
-t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda
-(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k
-u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda
-(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1
-t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1
-t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef
-i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S
-i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T
-(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2:
-T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0)
-v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0:
-T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq
-T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3
-t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
-u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0:
-K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
-\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _)
-\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0)
-\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T
-u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
-(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3:
-T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t)
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda
-(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1)))
-(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead
-k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda
-(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind
-T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda
-(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
-(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0
-(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2
-(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7)
-in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1
-H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3
-t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
-(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1
-t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1)
-(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T
-(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3:
-T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0
-H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2:
-T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0
-t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
-T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
-(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
-(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
-t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1
-t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda
-(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead
-k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u
-u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T
-(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2:
-T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2)
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0
-(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
-T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
-(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
-(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
-t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s
-k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1:
-K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2
-(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T
-(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s
-k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1
-t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s
-k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
-(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2:
-T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2)
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T
-(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2:
-T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2)
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3:
-T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0)
-v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5))
-H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda
-(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead
-k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
-(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0:
-T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4:
-(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
-k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
-(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
-v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
-t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
-\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _)
-\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T
-u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t:
-T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
-k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
-(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
-v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4
-t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t
-t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1
-(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1)))
-(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T
-t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3)))
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3))))
-(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in
-(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12
-k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T
-(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
-(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda
-(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
-(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
-i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead
-k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
-(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0
-(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda
-(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0
-v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3)))
-(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
-(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
-i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0
-H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))).
-
-theorem subst0_gen_lift_lt:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1)
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst0 i u t1 t2)))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d
-u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n:
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n))
-x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t:
-T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d))))
-in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n)
-t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S
-(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2:
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
-n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
-(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
-(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq
-T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
-(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
-(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))))
-(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0)
-O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda
-(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
-(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T
-(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O
-(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3)))
-(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
-n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
-T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
-(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
-h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
-(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
-u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n))
-H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2:
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
-n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4))))
-(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t)
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
-(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift
-h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0
-t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t
-t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0))
-(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i
-d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d))))
-in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus
-i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d))
-t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t)
-t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
-d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S
-(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h
-d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
-t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k
-(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S
-(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
-(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h
-(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
-T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d)))
-t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t)
-x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T
-x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T
-(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
-(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda
-(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d)))
-t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
-t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T
-(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h
-(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k
-(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1
-t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3:
-(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2)))
-(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d)))
-t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i
-d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
-(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
-T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda
-(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0)
-x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2
-T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3:
-T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i
-d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S
-(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus
-i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0))
-H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
-(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
-(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d))
-x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s
-k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h
-(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i
-u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T
-(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1))
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
-t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda
-(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind
-T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
-(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i
-d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d))
-(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i
-d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d))))
-(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0
-H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2)))
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
-(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq
-T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d
-u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
-(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda
-(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x
-(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d))
-t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
-d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
-(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda
-(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i
-d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d))
-(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7
-(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
-(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda
-(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k
-i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T
-(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S
-(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S
-(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
-x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
-t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T
-(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S
-(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind
-nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k
-(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus
-i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S
-(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2:
-T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead
-k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus
-i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T
-(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k
-x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u
-t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
-(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d)))
-(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0
-i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
-(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i
-H2))))))))))))) t1)).
-
-theorem subst0_gen_lift_false:
- \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
-(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u
-(lift h d t) x) \to (\forall (P: Prop).P)))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x:
-T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i
-(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
-Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda
-(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n))
-x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda
-(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in
-(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u:
-T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i:
-nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1:
-(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P
-(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda
-(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind
-(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda
-(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0:
-nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n
-H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n))
-(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d
-H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P
-(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n
-h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d
-h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h
-n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k:
-K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall
-(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h))
-\to ((subst0 i u (lift h d t0) x) \to (\forall (P:
-Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall
-(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to
-((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P:
-Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus
-d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P:
-Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2:
-T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1))
-(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k
-u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)))
-(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2:
-T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
-(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2:
-T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u
-(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
-(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda
-(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7:
-(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda
-(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
-(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2:
-T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
-(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k
-(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1)
-x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h))
-(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i
-(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7:
-(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d)
-t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d
-t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
-
-theorem subst0_gen_lift_ge:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h)
-i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
-T).(subst0 (minus i h) u t1 t2))))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h
-d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
-t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n:
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus
-d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0
-i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2
-T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i
-h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i:
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d
-(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
-n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
-(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
-(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq
-T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
-(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
-\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus
-d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
-T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
-(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
-T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
-(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S
-(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
-(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
-h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
-h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
-(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S
-(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d
-t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2))))
-(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S
-(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n)
-t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h
-d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u)
-(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n
-h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
-u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
-nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
-(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n
-h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d
-(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n
-d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i
-H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h)
-i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
-T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0:
-((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
-nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0
-t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda
-(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0))
-(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d)
-t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x
-(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t)
-u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda
-(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
-k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d
-t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
-(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0))))
-(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2:
-T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u
-(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
-(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u
-(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
-(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0
-(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda
-(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2:
-T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7:
-(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r
-T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
-(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u
-(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
-(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
-h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u
-(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0)))
-(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k
-d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4))
-(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2)))
-(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T
-(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0
-(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d
-t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda
-(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6:
-(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t)
-x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3)))
-(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T
-(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0
-(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d
-t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0)
-t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda
-(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1)
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2)
-(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
-(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3:
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t
-t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n:
-nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r
-d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1))
-(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))
-(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0
-(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t
-x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h))
-(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h)
-(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
-k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
-u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift
-h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda
-(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u
-(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0)
-x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq
-T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0)
-t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda
-(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T
-(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
-k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d)
-x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda
-(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t
-t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda
-(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t
-x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T
-(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead
-k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T
-(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda
-(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d
-(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d
-t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12
-\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9
-(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T
-(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal
-T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2
-H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d))
-x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind
-nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i
-H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u
-(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/fwd.ma".
-
-theorem subst0_refl:
- \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to
-(\forall (P: Prop).P))))
-\def
- \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d:
-nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n:
-nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort
-n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda
-(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef
-n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O
-u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O
-u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1
-P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0:
-T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P:
-Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u
-t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0
-d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T
-(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2:
-T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead
-k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T
-(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1)))
-(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T
-(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _)
-\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def
-(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6
-P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1)
-(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T
-(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2:
-T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k
-t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2)
-\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def
-(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s
-k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0
-t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2)))
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0
-x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1
-x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead
-_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2)
-\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T
-t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1
-t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u
-t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0
-t1 (THead k t0 t1) d H1)))))))))) t)).
-
-theorem subst0_lift_lt:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
-i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i
-(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall
-(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d
-t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda
-(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t:
-T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v))))
-(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0)))
-(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O
-v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t:
-T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S
-i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d
-v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r
-(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda
-(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_:
-(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall
-(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d
-u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt
-i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d)
-t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d
-(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t))
-(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d
-u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h
-d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k
-u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h
-d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1:
-((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0)
-(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda
-(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3
-\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s
-k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v)
-(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r
-T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0
-(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead
-k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h
-(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind
-nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v)
-(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s
-k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s
-k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d
-u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0))
-(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h
-d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d:
-nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v)
-(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda
-(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d:
-nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d
-(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda
-(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0))
-(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0:
-nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0
-t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1)
-(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t
-(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k
-d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift
-h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v)
-(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k
-d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s
-k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d)
-(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d
-(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0))
-(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
-
-theorem subst0_lift_ge:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
-(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0
-(plus i h) u (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n:
-nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le
-d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v:
-T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T
-(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift
-(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0
-(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda
-(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r
-nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0))
-O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S
-i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d
-(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d
-H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
-nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le
-d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t:
-T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T
-(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0
-h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift
-h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1)
-(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0
-h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2
-t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda
-(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0:
-nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d:
-nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d
-t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3
-\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0:
-nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1
-(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0)
-(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead
-k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda
-(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t))
-(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k
-d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k
-u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d)))))))))))))
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda
-(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to
-(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k:
-K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0
-t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k
-i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le
-d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n:
-nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0)
-(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T
-(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0
-h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift
-h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1)
-(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0
-h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d
-i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead
-k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
-
-theorem subst0_lift_ge_S:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
-i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d
-t1) (lift (S O) d t2))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat
-(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d
-t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O)
-i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O))
-(plus_sym i (S O)))))))))).
-
-theorem subst0_lift_ge_s:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
-i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s
-(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda
-(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/props.ma".
-
-theorem subst0_subst0:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
-u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
-T).(subst0 (S (plus i j)) u t t2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
-(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
-T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n
-u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3)))))))))))
-(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
-(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i))
-(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
-(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t:
-T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t
-(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v
-u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i))
-(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0:
-T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
-u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0:
-nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t))
-(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t:
-T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0:
-nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i
-u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda
-(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0
-i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1
-x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0:
-T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i))
-u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst
-u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k:
-K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i:
-nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1:
-T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda
-(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
-i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0:
-T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda
-(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
-i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t))
-(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x:
-T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0
-(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n:
-nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in
-(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n
-u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T
-(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S
-(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3
-u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0
-H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
-(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3:
-T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda
-(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t
-u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
-(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u:
-T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0
-(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t
-t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4:
-(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t))
-(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t:
-T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
-t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0
-x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda
-(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))
-(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t:
-T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda
-(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0
-u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0
-(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def
-(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9
-(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t:
-T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
-t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5)
-(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4)))))
-(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))).
-
-theorem subst0_subst0_back:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
-u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
-T).(subst0 (S (plus i j)) u t2 t)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
-(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
-T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n
-u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4)))))))))))
-(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
-(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i))
-(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
-(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t:
-T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift
-(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0
-(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0
-(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda
-(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1:
-((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to
-(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus
-i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3:
-T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v
-u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0:
-T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3
-(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t)
-t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0
-(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k
-u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0))
-(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0
-i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v:
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0
-(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall
-(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1
-t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda
-(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2:
-(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t))
-(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t:
-T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0
-(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3
-x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def
-(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4
-(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k
-(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i)))
-(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u
-t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead
-k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6
-u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1:
-T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
-u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0:
-nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t))
-(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k:
-K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0
-t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0:
-nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t))
-(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3:
-T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v
-u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t:
-T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3
-(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3)
-t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6:
-(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i
-u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda
-(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0
-i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1
-x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r
-nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus
-i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0
-i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k
-(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0)
-t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0
-x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i))
-H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2
-t1 t2 H))))).
-
-theorem subst0_trans:
- \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0
-i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1
-t3)))))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to
-(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3:
-T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false
-v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0
-(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1:
-T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1:
-((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda
-(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k
-u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t)))
-(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3
-(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
-T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda
-(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3:
-T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3
-t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3)
-(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0
-i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead
-k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda
-(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4:
-T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k
-u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k
-u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda
-(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0:
-T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3
-H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T
-t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3)))
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T
-(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
-T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5:
-(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T
-(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0))
-(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3))
-(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0:
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0
-(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0
-t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4:
-T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda
-(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))
-(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s
-k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
-(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2)))
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0
-(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2
-t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq
-T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0
-(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x
-t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda
-(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0)
-t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u
-t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5:
-T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))
-(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4
-(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k
-u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3
-i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0
-t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3)
-t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0
-x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0
-x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3)
-t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3))
-(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda
-(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1
-u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0
-u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2:
-(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0)
-v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4:
-(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4
-(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda
-(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3
-t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
-t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1
-t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3)))
-(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4
-(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead
-k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda
-(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0
-i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4
-H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5)))
-(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5:
-T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))
-(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4
-(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead
-k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1
-u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda
-(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda
-(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0
-i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))
-(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda
-(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t:
-T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0
-x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0
-H4))))))))))))))) i v t1 t2 H))))).
-
-theorem subst0_confluence_neq:
- \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
-nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
-(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
-(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t))))))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
-nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n:
-nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4:
-T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq
-nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5:
-T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda
-(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2
-(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq
-T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v)
-t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda
-(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n:
-nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda
-(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda
-(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in
-False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift
-(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with [])
-in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v:
-T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0
-i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2:
-nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t:
-T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda
-(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2:
-nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq
-nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t)))
-(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2
-(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T
-(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4:
-T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead
-k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T
-(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0
-u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4:
-T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t)
-t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq
-T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k
-x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t)
-t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3:
-T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda
-(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
-k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda
-(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k
-u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t)
-(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2
-H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead
-k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda
-(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t
-t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3:
-T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0
-x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x)
-(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4))
-(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0
-i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3))
-(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k))
-t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq
-T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0
-u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t
-t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4
-t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3:
-T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3)))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0
-x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t
-x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4:
-T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4))))
-(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i
-v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda
-(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8:
-(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda
-(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
-k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7)
-(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4))
-(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda
-(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0:
-(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2:
-T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2))
-\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k
-i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2:
-T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3)
-t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq
-T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda
-(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3
-t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
-t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t:
-T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t)))
-(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda
-(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k
-u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0
-i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
-T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u
-x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5:
-T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
-(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t:
-T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2
-k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T
-(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2)
-u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda
-(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2
-(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
-T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2
-t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5:
-T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
-(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0
-(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t))
-(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7:
-(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2
-T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i
-v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u)
-(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t:
-T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq
-nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k
-i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s
-k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7:
-(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4))
-(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k
-u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda
-(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k
-u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u
-x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1)
-(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5))
-(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k
-i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t:
-T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0
-x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda
-(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2
-(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k
-x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0)))))
-(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t))
-(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to
-False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_:
-(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj
-k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k
-i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3
-t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2:
-T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1:
-((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2)
-\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t))
-(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2:
-T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3:
-((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4)
-\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3
-t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4:
-T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k
-u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4:
-T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T
-(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2)
-u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4
-t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
-T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t:
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
-(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda
-(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k
-u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t:
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
-(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0
-i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5:
-T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
-(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x
-t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t:
-T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2
-u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t:
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x
-t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0
-i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T
-(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2)
-u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda
-(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3
-(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
-T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3
-t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5:
-T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
-(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0
-(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t))
-(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda
-(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x
-x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda
-(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3
-i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8
-(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0
-(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0:
-T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x
-x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11))))))
-(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i
-i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4:
-T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_:
-T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2)
-u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4
-(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4)))
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda
-(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0
-x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3
-t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5:
-T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
-(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v
-x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t:
-T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2
-u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t:
-T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T
-(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v
-(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3
-x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t:
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0
-x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both
-v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t:
-T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq
-nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k
-i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat
-(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda
-(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2
-H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2
-H4)))))))))))))))))) i1 u1 t0 t1 H))))).
-
-theorem subst0_confluence_eq:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
-i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2)
-(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t)))
-(subst0 i u t1 t2) (subst0 i u t2 t1))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to
-(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5:
-T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda
-(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef
-i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T
-(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v)
-t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2)
-(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda
-(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2)
-(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t:
-T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2
-(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v
-t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda
-(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2:
-T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0
-i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0
-i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda
-(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T
-t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda
-(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t
-t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2)
-(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
-T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
-(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3
-t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq
-T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead
-k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
-(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
-(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x
-t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda
-(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v
-(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v
-(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x)
-(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x
-t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead
-k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
-(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k
-x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2
-x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t))
-(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4:
-T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x
-t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x
-t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))
-(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t)
-(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead
-k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3))
-(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0
-v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t)
-(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3))
-(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t)
-(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0:
-T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x
-x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3:
-T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x
-t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x
-t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t)
-t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t)
-(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6))
-(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x
-t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
-T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x
-t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t
-k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t)
-(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3))
-(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t)
-(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x
-i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3:
-T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t
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-(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead
-k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2
-i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5))
-(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))).
-
-theorem subst0_confluence_lift:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
-i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i
-t2)) \to (eq T t1 t2)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0
-i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i
-t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t:
-T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S
-O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2)
-(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def
-(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i
-H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t))
-(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t:
-T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O)
-i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i
-t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1)
-x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S
-O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O))
-(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift
-(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1)
-(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n))
-(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2)))
-(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i
-t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i)
-(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O)
-i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2)))
-(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/defs.ma".
-
-include "LambdaDelta-1/lift/props.ma".
-
-include "LambdaDelta-1/lift/tlt.ma".
-
-theorem subst0_weight_le:
- \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
-u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
-(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
-(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1))))))))))
-(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
-((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
-(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift
-(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O
-v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2:
-T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
-u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda
-(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
-k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
-(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
-(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
-((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
-m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S
-(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
-(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
-(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
-(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
-g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
-g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
-(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
-H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
-(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
-(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
-t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
-t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd
-g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f:
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
-(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2)
-(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f
-g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g
-H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
-i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
-u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1)
-(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
-H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
-t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g
-t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g:
-((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
-f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2))
-(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda
-(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i)))
-\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
-T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
-(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
-t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
-(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
-i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
-u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
-t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S
-(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
-(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
-(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
-(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
-u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
-nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S
-i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f))))))))))))))))
-(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
-(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
-t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
-m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
-i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
-(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0)
-(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
-(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
-g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
-(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
-(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
-t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g
-t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
-(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0)
-(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
-t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O)
-t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g
-O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map
-f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O)
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b))
-(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
-(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g
-m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0
-t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
-i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g
-u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0)
-(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2
-H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall
-(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f
-m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le
-(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda
-(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to
-(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s
-k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f:
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
-(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map
-f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b:
-B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s
-(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat
-\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
-(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f
-t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g:
-((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
-f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2))
-(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
-(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f:
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
-m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
-u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
-t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
-(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f
-g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1)))
-(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1))
-(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5
-(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v))
-(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
-((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
-i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
-(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
-(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2)
-(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
-g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O
-(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
-nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v))
-(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
-t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
-\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5:
-(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
-(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
-t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
-t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
-(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f
-(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b))
-(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1
-t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
-(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g
-t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5:
-(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2)
-(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus
-(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1
-f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
-
-theorem subst0_weight_lt:
- \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
-u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
-(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
-(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1))))))))))
-(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
-((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
-(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v:
-T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i
-v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda
-(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
-k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
-(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
-(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
-((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
-m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S
-(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
-(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
-(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
-(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
-g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
-g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S
-(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
-H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
-(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
-(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
-t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
-O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
-(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd
-f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f:
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
-(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f
-u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
-(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n
-(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O
-O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
-i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
-u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1)
-(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
-H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
-t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g
-t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g:
-((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
-f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2))
-(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda
-(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i)))
-\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
-T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
-(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
-t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
-(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
-i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
-u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
-t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
-(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
-(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
-(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
-(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
-u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
-nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S
-i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f))))))))))))))))
-(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
-(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f
-t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
-m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
-i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
-(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f
-u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
-(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
-g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
-(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
-(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
-t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
-t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
-(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0)
-(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
-t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
-O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd
-g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
-(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
-f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
-t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
-(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g
-t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
-(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
-f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1))
-(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
-(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k))
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
-(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall
-(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt
-(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map
-g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1:
-T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt
-(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
-k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind
-(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v
-t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2)
-(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
-\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
-(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2))
-(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f:
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
-m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
-u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
-t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
-(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1
-f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2)))
-(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S
-(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2)
-(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O
-v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f
-u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i)
-f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v
-t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
-t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
-(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
-(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
-t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
-t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
-(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m)
-(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f
-(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
-((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
-i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
-(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
-(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2)
-(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
-g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m)
-(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O)
-m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g
-i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v
-(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m))))
-\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2)
-(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat
-\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
-(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
-f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
-(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
-(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t
-z H))))).
-
-theorem subst0_tlt_head:
- \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt
-(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
-z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
-(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map
-(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n
-(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda
-(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m)))
-(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n:
-nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda
-(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_:
-nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u)
-(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda
-(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda
-(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
-
-theorem subst0_tlt:
- \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z
-(THead (Bind Abbr) u t)))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
-z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx
-(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/defs.ma".
-
-inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def
-| subst1_refl: subst1 i v t1 t1
-| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst1/defs.ma".
-
-include "LambdaDelta-1/subst0/props.ma".
-
-theorem subst1_gen_sort:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
-i v (TSort n) x) \to (eq T x (TSort n))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T
-t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0
-i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x
-H))))).
-
-theorem subst1_gen_lref:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
-i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift
-(S n) O v))))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or
-(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl
-(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O
-v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v
-(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
-(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq
-nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
-(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i)
-(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x
-H))))).
-
-theorem subst1_gen_head:
- \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
-(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst1 (s k i) v t1 t2))))))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1)
-x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1
-t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1
-t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal
-T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda
-(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T
-(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
-u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
-T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2
-(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda
-(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda
-(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1
-x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1
-x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3:
-T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
-T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0:
-T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v
-t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1)
-(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda
-(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1
-i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0
-x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4)))))))
-H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
-
-theorem subst1_gen_lift_lt:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1)
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst1 i u t1 t2)))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S
-(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1)
-(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
-(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2:
-T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1))
-(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u)
-(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h
-(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1
-t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d))
-x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T
-t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1
-(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x
-H))))))).
-
-theorem subst1_gen_lift_eq:
- \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
-(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u
-(lift h d t) x) \to (eq T x (lift h d t))))))))))
-\def
- \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d
-h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t)
-(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda
-(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t
-u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
-
-theorem subst1_gen_lift_ge:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h)
-i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
-T).(subst1 (minus i h) u t1 t2))))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1)
-x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda
-(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2:
-T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
-h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1
-(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2:
-T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3:
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3))
-(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1
-(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d
-x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3:
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0
-H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h
-d H1 H0)))) x H)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst1/defs.ma".
-
-include "LambdaDelta-1/subst0/props.ma".
-
-theorem subst1_head:
- \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1
-i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s
-k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2))))))))))
-\def
- \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k:
-K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i
-v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k
-i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t)))
-(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k
-i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k
-v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
-t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1
-(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead
-k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1)
-(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k
-i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both
-v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
-
-theorem subst1_lift_lt:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1
-i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i
-(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d:
-nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u)
-(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i
-d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d
-t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d:
-nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h
-(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d
-H1 h))))))) t2 H))))).
-
-theorem subst1_lift_ge:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
-(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1
-(plus i h) u (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t:
-T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h
-d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u
-(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda
-(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1)
-(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
-
-theorem subst1_ex:
- \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2:
-T).(subst1 d u t1 (lift (S O) d t2))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex
-T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n:
-nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n)
-(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d
-u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n))
-(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d
-(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda
-(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O)
-d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n)
-t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S
-O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T
-(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T
-(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u)
-(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n)
-t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S
-O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d
-H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n)
-(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t:
-T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef
-(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t:
-T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift
-(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T
-(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d:
-nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2:
-T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u
-(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u
-t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
-(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex
-T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda
-(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d)
-x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d
-t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k
-d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t
-(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k
-x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
-
-theorem subst1_lift_S:
- \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i
-(TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
-\def
- \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h:
-nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i
-t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_:
-(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift
-(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef
-h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n))
-(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S
-i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H:
-(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n))
-(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n)
-(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T
-(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i
-(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0))
-(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0))))
-(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0:
-nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef
-h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T
-(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n))))
-(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef
-n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h)
-(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n
-(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0:
-nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O
-(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n
-(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n))
-(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n))
-(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h)
-(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n))
-(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt
-n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T
-(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i
-(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i
-(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n
-(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S
-(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i)
-H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i:
-nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t)
-(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i:
-nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i)
-t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1:
-(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
-t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0))))
-(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1:
-T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
-t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i
-h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S
-(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift
-(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k
-(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i))
-(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst1/fwd.ma".
-
-include "LambdaDelta-1/subst0/subst0.ma".
-
-theorem subst1_subst1:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
-u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
-T).(subst1 (S (plus i j)) u t t2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
-(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
-T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda
-(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
-t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
-(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
-(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl
-(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1
-t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1
-i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_:
-T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S
-(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1
-y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0:
-T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3)))))
-(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda
-(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
-t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t:
-T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0)
-(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3:
-(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0
-(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t:
-T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T
-(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
-t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7:
-(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1
-t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1
-x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j
-H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))).
-
-theorem subst1_subst1_back:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
-u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
-T).(subst1 (S (plus i j)) u t2 t)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
-(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
-T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda
-(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t
-t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
-(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
-(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl
-(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1
-t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1
-i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0:
-T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0))))
-(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S
-(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i
-j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T
-(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u
-t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S
-(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1
-x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t:
-T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x
-(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4)))))
-(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))).
-
-theorem subst1_trans:
- \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1
-i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1
-t3)))))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3:
-T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda
-(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1
-t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3
-(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0:
-T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans
-t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))).
-
-theorem subst1_confluence_neq:
- \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
-nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
-(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
-(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t))))))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
-nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t:
-T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2)
-\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3))
-(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2:
-T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not
-(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda
-(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2:
-T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2:
-T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not
-(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4:
-T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T
-(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2
-(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4:
-T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1
-u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1
-i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda
-(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T
-(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x
-(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4)))))
-(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2
-H2))))) t3 H1)))))))) t1 H))))).
-
-theorem subst1_confluence_eq:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
-i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t:
-T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t)))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2:
-T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3))
-(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0:
-(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda
-(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2:
-T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i
-u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1
-i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t:
-T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u
-t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u
-t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t))
-(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4)
-(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)))
-(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda
-(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2
-T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2
-(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T
-(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2
-t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i
-u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i
-u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5:
-(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda
-(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4
-x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t:
-T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u
-t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2
-t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1
-i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4)))
-(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))).
-
-theorem subst1_confluence_lift:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
-i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i
-t2)) \to (eq T t1 t2)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1)
-(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1
-i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0:
-(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i
-t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1
-t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda
-(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda
-(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4
-\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u
-(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
-(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O)))
-H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1:
-(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3:
-T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2
-(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T
-(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1
-t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0
-(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6:
-(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t:
-T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in
-(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r
-nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i
-(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6:
-(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def
-(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in
-(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5)))
-H3))))))) y H0))) H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/subst0/tlt.ma".
-
-include "LambdaDelta-1/subst/props.ma".
-
-include "LambdaDelta-1/sty1/cnt.ma".
-
-include "LambdaDelta-1/ex0/props.ma".
-
-include "LambdaDelta-1/wcpr0/fwd.ma".
-
-include "LambdaDelta-1/pr3/wcpr0.ma".
-
-include "LambdaDelta-1/ex2/props.ma".
-
-include "LambdaDelta-1/ex1/props.ma".
-
-include "LambdaDelta-1/ty3/sty0.ma".
-
-include "LambdaDelta-1/csubt/csuba.ma".
-
-include "LambdaDelta-1/ty3/fwd_nf2.ma".
-
-include "LambdaDelta-1/ty3/nf2.ma".
-
-include "LambdaDelta-1/wf3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-inductive TList: Set \def
-| TNil: TList
-| TCons: T \to (TList \to TList).
-
-definition THeads:
- K \to (TList \to (T \to T))
-\def
- let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t:
-T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u
-(THeads k ul t))])) in THeads.
-
-definition TApp:
- TList \to (T \to TList)
-\def
- let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts
-with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t
-(TApp ts0 v))])) in TApp.
-
-definition tslen:
- TList \to nat
-\def
- let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O
-| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen.
-
-definition tslt:
- TList \to (TList \to Prop)
-\def
- \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tlist/defs.ma".
-
-theorem tslt_wf__q_ind:
- \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
-\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
-\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
-ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
-(tslen ts)))))).
-
-theorem tslt_wf_ind:
- \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
-TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
-TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
-Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
-(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
-TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
-nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
-(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
-m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
-\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
-(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
-H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
-ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
-
-theorem theads_tapp:
- \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T
-(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
-TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads
-k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda
-(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k
-v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0
-(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads
-k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
-
-theorem tcons_tapp_ex:
- \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
-TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
-(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
-\def
- \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
-TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
-ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
-ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
-TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
-(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
-TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
-T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
-(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
-t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
-ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
-(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
-t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
-(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
-TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
-(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
-(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
-nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
-TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
-t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
-(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
-nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
-t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
-nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
-TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
-(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
-(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
-(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
-
-theorem tlist_ind_rev:
- \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
-TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
-TList).(P ts))))
-\def
- \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
-((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
-t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
-(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
-TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
-TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
-TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
-\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
-\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
-(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
-t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
-(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
-T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
-(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
-t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
-(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
-H4))))) H3))))))) ts2)) ts)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/T/defs.ma".
-
-definition wadd:
- ((nat \to nat)) \to (nat \to (nat \to nat))
-\def
- \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n
-with [O \Rightarrow w | (S m) \Rightarrow (f m)]))).
-
-definition weight_map:
- ((nat \to nat)) \to (T \to nat)
-\def
- let rec weight_map (f: ((nat \to nat))) (t: T) on t: nat \def (match t with
-[(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0)
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr
-\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f
-u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f
-O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O)
-t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f
-t0)))])]) in weight_map.
-
-definition weight:
- T \to nat
-\def
- weight_map (\lambda (_: nat).O).
-
-definition tlt:
- T \to (T \to Prop)
-\def
- \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/tlt/defs.ma".
-
-theorem wadd_le:
- \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
-nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to
-(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
-\def
- \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
-((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
-nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
-nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le
-(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
-
-theorem wadd_lt:
- \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
-nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to
-(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
-\def
- \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
-((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
-nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
-nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0))
-(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0)))
-n))))))).
-
-theorem wadd_O:
- \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_:
-nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat
-(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n).
-
-theorem weight_le:
- \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t)
-(weight_map g t)))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda
-(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall
-(n0: nat).(le (f n0) (g n0))))).(le_n (weight_map g (TSort n))))))) (\lambda
-(n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
-(H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) (\lambda (k:
-K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1:
-T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1))))))
-\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1))
-(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0:
-B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0)
-(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0)
-(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus
-(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus
-(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr
-\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g
-t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g
-O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O)
-t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
-(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g
-t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
-(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus
-(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1))
-(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S
-(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g
-H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0)))
-(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0))
-(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n))))))))))))
-(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g:
-((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f
-t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f:
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
-(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
-(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1))
-(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map
-f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1)
-(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O
-(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
-(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g
-t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
-(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0)
-(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0)
-(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f
-O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n))))))))))))
-b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n))))
-\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
-(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g
-t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus
-(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g
-t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1)
-(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t).
-
-theorem weight_eq:
- \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f
-t) (weight_map g t)))))
-\def
- \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym
-(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n:
-nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n)
-(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0:
-nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))).
-
-theorem weight_add_O:
- \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t)
-(weight_map (\lambda (_: nat).O) t))
-\def
- \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_:
-nat).O) (\lambda (n: nat).(wadd_O n))).
-
-theorem weight_add_S:
- \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O)
-O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
-\def
- \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O)
-(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_:
-nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m
-(le_O_n m)) n)))).
-
-theorem tlt_trans:
- \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to
-(tlt u t)))))
-\def
- \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u)
-(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u)
-(weight v) (weight t) H H0))))).
-
-theorem tlt_head_sx:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt
-(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead
-k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
-(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
-(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
-(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_:
-nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_:
-nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map
-(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map
-(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O)
-u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda
-(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O)
-t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S
-(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u)
-(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_:
-nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k).
-
-theorem tlt_head_dx:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt
-(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead
-k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
-(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
-(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S
-(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_:
-nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_:
-nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S
-(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u)
-(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O)
-u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
-(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd
-(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda
-(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t
-(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t)
-(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t)))))))
-(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_:
-nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus
-(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_:
-nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_:
-nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map
-(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t)
-(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map
-(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_:
-nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S
-(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u)
-(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_:
-nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda
-(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u:
-T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus
-(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))
-(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_:
-nat).O) t)))))) k).
-
-theorem tlt_wf__q_ind:
- \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to
-Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0
-t))))) P n))) \to (\forall (t: T).(P t)))
-\def
- let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
-T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t)
-n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight
-t)))))).
-
-theorem tlt_wf_ind:
- \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t)
-\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t)))
-\def
- let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
-T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
-Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v)
-(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind
-(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0:
-T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
-\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat
-(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
-(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P
-t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt
-(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight
-v))))))))))))) t)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/pr3_props.ma".
-
-include "LambdaDelta-1/arity/pr3.ma".
-
-include "LambdaDelta-1/asucc/fwd.ma".
-
-theorem ty3_arity:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
-t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity
-g c t2 (asucc g a1))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
-(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g
-c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity
-g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u:
-T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
-a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda
-(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1)))
-(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3
-(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_:
-(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1:
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g
-a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10:
-(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0:
-T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1:
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1))))
-(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3
-x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
-g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g
-(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12
-g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11)))))
-H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A
-(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0
-(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort
-g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
-(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
-a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
-(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
-c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g
-a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g
-d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1))
-(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g
-c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr
-c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
-(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
-(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
-a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
-(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
-c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
-a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d
-t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind
-A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g
-c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
-a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A
-(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0
-(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g
-d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g
-d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7))))
-H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity
-g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b:
-B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
-u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b)
-u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g
-a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1))
-(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
-g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b)
-u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u
-x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A
-(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1:
-A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1:
-A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead
-(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead
-c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc
-g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A
-(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0
-(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4)
-(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g
-x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to
-((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1:
-A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0
-(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0
-(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4
-(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u
-t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1)))
-x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr
-not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g
-(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind
-Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead
-(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4)
-(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5
-(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead
-x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1)
-H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead
-x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3
-x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g
-x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1))
-(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0
-(arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t3 x0
-H12) (arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t4
-(asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda
-(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
-(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0
-u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v
-(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v
-a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g
-a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1))
-(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
-g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
-Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda
-(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def
-H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity
-g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
-c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
-Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda
-(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t)
-(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in
-(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1
-a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
-(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))
-(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda
-(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g
-a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0)
-(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13:
-(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0)
-(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in
-(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A
-x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
-a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
-t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1
-x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2
-(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17)
-in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1
-x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
-a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
-t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1
-(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc
-g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc
-g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10)))))
-H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1:
-A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
-a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A
-(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
-a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1))
-(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
-g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat
-Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3
-x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A
-(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
-a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1))
-(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1))))
-(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0
-t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat
-Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4)
-(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g
-x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0
-(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7)))))
-H4)))))))))) c t1 t2 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/arity.ma".
-
-include "LambdaDelta-1/sc3/arity.ma".
-
-theorem ty3_predicative:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u:
-T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P:
-Prop).P)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u:
-T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u
-v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2:
-T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_:
-T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v
-x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def
-(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def
-(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind
-A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1:
-A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c
-(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8
-\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1:
-A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
-A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
-(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda
-(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g
-x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def
-(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9)
-in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead
-x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g
-Abst c v t u H1)))))))))).
-
-theorem ty3_repellent:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1:
-T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead
-c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P:
-Prop).P)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1:
-T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2:
-T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O
-u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0:
-T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x:
-T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3
-\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop
-(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3
-(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2
-t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O)
-O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5
-(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind
-Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda
-(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g
-(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1:
-A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity
-g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def
-(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in
-(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda
-(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity
-g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g
-x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g
-c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1)
-(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2)
-H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w)
-t (lift (S O) O u2) H0))))))))))).
-
-theorem ty3_acyclic:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
-u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def
-(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in
-(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
-(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g
-c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x
-(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
-
-theorem ty3_sn3:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
-u) \to (sn3 c t)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in
-(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u
-(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t
-x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t
-x H1))))) H0))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pc3/dec.ma".
-
-include "LambdaDelta-1/getl/flt.ma".
-
-include "LambdaDelta-1/getl/dec.ma".
-
-theorem ty3_inference:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2:
-T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0:
-C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2:
-T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind
-(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or
-(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4)
-\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall
-(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_:
-((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T
-(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
-False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)))
-(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda
-(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n)))))
-(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3
-c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
-T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let
-H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
-(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d)
-\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n)
-t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1:
-(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead
-e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda
-(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g
-c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)))
-(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2
-(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2))
-in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g
-x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
-(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T
-(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3:
-T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g
-x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or
-(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2
-(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr)
-x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall
-(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3:
-T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x
-H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T
-(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef
-n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))
-(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2
-(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2
-(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))
-(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t)
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2
-(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
-n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
-T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind
-C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O
-t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_:
-(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind
-Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0
-(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4)
-(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10))
-in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
-ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat
-_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0
-(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind False
-H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_:
-(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind
-Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0
-(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4)
-(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10))
-in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
-ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
-\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat
-_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0
-(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind False
-H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda
-(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T
-(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef
-n) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n)
-t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
-T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False
-(\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
-T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift
-(S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr)
-x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind
-x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4)
-(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in
-(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
-C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0
-(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal
-C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0
-(Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2)
-n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda
-(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2
-| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind
-Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr)
-x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let
-H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t)))
-H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5))
-H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead
-c0 (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda
-(c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H19)))))))) H12))
-H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7:
-(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst)
-x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind
-x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4)
-(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in
-(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
-C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0
-(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal
-C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0
-(Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2)
-n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda
-(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2
-| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind
-Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst)
-x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let
-H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t)))
-H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5))
-H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S
-n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0:
-C).(getl n c2 (CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def
-(eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5
-H20))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5))))))
-H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P:
-Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
-(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (\lambda (t3:
-T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
-T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t))))) False (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
-T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t)
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_:
-(pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr)
-x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5
-False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
-T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_:
-(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst)
-x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5
-False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1
-t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4:
-T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g
-c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0:
-T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to
-(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3
-t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3)))
-(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall
-(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T
-(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
-False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3:
-T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1
-t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T
-(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall
-(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T
-(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to
-False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in
-(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2
-t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0)
-t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False)))
-(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda
-(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b)
-t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to
-False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2
-(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T
-(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3
-g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g
-c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t
-t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2
-(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t)
-t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)))
-(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda
-(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T
-(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3
-g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3
-g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b
-t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b)
-t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead
-(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3)
-\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0)
-t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b)
-t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4:
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda
-(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7
-x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4:
-((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda
-(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g
-c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_:
-T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3
-g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0
-t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead
-(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead
-c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5))))))
-H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt
-c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
-t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda
-(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t
-t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3
-g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead
-(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3)
-\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3
-c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3
-t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3
-c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3:
-T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T
-(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)))
-(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)))
-(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0
-(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
-c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda
-(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3:
-T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0:
-T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)))
-(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)))
-(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3:
-T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
-t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
-False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def
-(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def
-H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u))
-(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall
-(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3:
-T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def
-(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0
-x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u:
-T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u:
-T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_:
-T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2
-v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or
-(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2
-T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u))))
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1)))
-(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda
-(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3
-c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2
-(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3
-v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3:
-T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3
-g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda
-(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23
-\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24
-\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1))
-H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl)
-t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
-False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))
-(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred
-g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0
-x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0
-(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3:
-T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3
-g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda
-(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda
-(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5:
-T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5))
-t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22:
-(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0
-\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5
-(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5)
-H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3
-(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3
-H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2
-t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to
-False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
-t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
-False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0)
-t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat
-Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3
-g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2
-t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead
-(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0
-(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind
-Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5))
-(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex
-T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3:
-T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T
-(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind
-Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind
-Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t
-(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst)
-x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t
-t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt
-c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1
-t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def
-(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3:
-T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T
-(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)))
-(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
-(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0
-(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
-c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda
-(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2
-(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3:
-T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0:
-T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)))
-(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
-(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2
-x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0
-t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0)
-t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)))
-(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2
-(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast)
-t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat
-Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t
-x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to
-False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0)
-t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))
-(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0)
-t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3))
-(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False
-(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda
-(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def
-(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15
-x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g
-c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda
-(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3:
-T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda
-(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0
-t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_:
-(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0
-t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3
-H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to
-False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t
-t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to
-False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0)
-t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3))
-(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False
-(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda
-(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4:
-T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5
-x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6))))))
-H4))) f H2))) k H1))))))) t2))) c t1))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/G/defs.ma".
-
-include "LambdaDelta-1/pc3/defs.ma".
-
-inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def
-| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t)
-\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to
-(ty3 g c u t2))))))))
-| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next
-g m))))
-| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u:
-T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to
-(ty3 g c (TLRef n) (lift (S n) O t))))))))
-| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u:
-T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to
-(ty3 g c (TLRef n) (lift (S n) O u))))))))
-| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to
-(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b)
-u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))
-| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to
-(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3
-g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u
-t)))))))))
-| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2)
-\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1)
-(THead (Flat Cast) t0 t2))))))).
-
-inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
-| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c
-TNil u)))
-| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts:
-TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/props.ma".
-
-include "LambdaDelta-1/pc3/fsubst0.ma".
-
-include "LambdaDelta-1/getl/getl.ma".
-
-theorem ty3_fsubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
-t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind
-Abbr) u)) \to (ty3 g c2 t2 t))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
-(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
-(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3:
-T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind
-Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2:
-T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2
-c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2
-t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u
-t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2:
-C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c
-(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c
-t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
-T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
-\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u
-t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
-u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6)
-e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda
-(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2
-t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u
-(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3
-(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6:
-(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c
-c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr)
-u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8)
-t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c
-t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4
-H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i:
-nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0
-i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0:
-C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to
-(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i
-u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr)
-u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m))))))))
-(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda
-(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3:
-T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_:
-(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind
-Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g
-m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda
-(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i:
-nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2
-t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
-t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda
-(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c
-(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead
-e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3:
-T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5:
-(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S
-n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda
-(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4:
-T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0:
-nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C
-(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind
-Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0)
-H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d
-(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal
-C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u)
-(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e
-(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C
-e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let
-H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4)))
-H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift
-(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14))
-u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda
-(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5:
-(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift
-(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c
-c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind
-Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b)
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0))
-(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0
-H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
-u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
-T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n)
-(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1
-(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda
-(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow
-d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind
-x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
-x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
-(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
-(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14)
-in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind
-x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n
-c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus
-i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind
-Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i
-(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n
-H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6))
-in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd
-(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind
-Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda
-(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0:
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3
-(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1
-x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def
-(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14
-\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d
-(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr
-x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3:
-T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r
-C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let
-H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u)))
-H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0:
-nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0)))
-(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
-c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n
-n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr
-g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n))
-u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n
-(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr)
-x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13))
-H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
-Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0))
-(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
-(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
-x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11:
-(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0
-x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def
-(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15
-\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d
-(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr
-x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3:
-T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C
-x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let
-H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4)))
-H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0:
-nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0)))
-(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
-c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n
-n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr
-g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S
-n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S
-n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S
-(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))
-H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u
-(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1)))))))
-(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3:
-C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c
-(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0))
-(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq
-T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3
-g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0:
-nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def
-(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11
-\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0
-(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e
-(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono
-c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13
-\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d
-(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
-n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15
-\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11
-d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d
-(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4:
-T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3
-(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n)
-(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d
-(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i
-n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
-Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2:
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0:
-T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n)
-c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3:
-T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3
-(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n)
-t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr)
-u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S
-n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O
-u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n)
-O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
-(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
-(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c
-(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
-Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S
-n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n
-H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e:
-C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3
-(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def
-(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind
-(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
-T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n))
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S
-n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3
-d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
-(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n)
-(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11:
-(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead
-x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _
-t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in
-(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
-(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14)
-in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind
-x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n
-c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus
-i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind
-Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i
-(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n
-H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6))
-in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3
-O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst
-g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n))
-u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus
-i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u
-(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n)))
-(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u
-u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq
-C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
-Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
-u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda
-(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind
-x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind
-Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda
-(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst)
-u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
-x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let
-H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3)))
-H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i
-(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b:
-B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind
-nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e
-(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3
-(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead
-x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n)))
-(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0
-x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back
-(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e
-(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus
-i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
-n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S
-n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1
-(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda
-(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S
-n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in
-((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
-\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in
-((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3]))
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B
-Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda
-(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def
-(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d
-H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2
-(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n)
-(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr)
-u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n
-i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20
-(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in
-(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2
-(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e
-(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
-d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
-(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4
-n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus
-i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e
-(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
-d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
-(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop
-Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n))
-u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e
-(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n)))
-d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e
-(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8))
-H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c
-c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda
-(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0
-c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
-u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift
-(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O
-u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n)
-O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
-(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0:
-nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind
-Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0)
-(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
-(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in
-C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
-Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S
-n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n
-H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda
-(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i:
-nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2
-t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
-t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2:
-(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall
-(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u)
-t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0:
-T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead
-(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
-\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5:
-(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i
-c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead
-(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6:
-T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
-u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c
-t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5
-(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i
-u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8:
-(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T
-(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3)))
-(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c
-(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10:
-(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead
-c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3))
-(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g
-c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0
-H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0
-c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b)
-x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x)
-(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind
-Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3)
-(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd
-i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3
-(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0
-(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead
-c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
-H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead
-(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
-t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda
-(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3))
-(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9:
-(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda
-(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g
-(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u
-t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3
-x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x
-(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3
-(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b)
-u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5
-H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T
-t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
-u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c
-t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq
-T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda
-(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
-(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
-T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead
-(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3
-x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c
-(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_:
-(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3)
-(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1)
-(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c
-u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S
-i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind
-b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))
-(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead
-(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead
-(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b)))
-e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c
-(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead
-c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3
-H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5))))))
-(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda
-(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g
-(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b)
-u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3
-x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2
-t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind
-b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g
-(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2
-(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u)
-(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind
-Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead
-(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c
-c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
-u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2)))
-(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead
-(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
-T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead
-(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind
-b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
-T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))
-(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5
-(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead
-(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind
-T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead
-(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g
-(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3
-(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3))
-(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T
-(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind
-b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead
-c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u
-x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3
-x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i
-u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead
-c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3
-(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b)
-i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3)
-(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind
-b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3)
-(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead
-c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i)
-u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u
-x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))))))
-(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3
-(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0
-(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0
-H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9))))
-H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6)))
-(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6:
-T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i)
-u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9:
-(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2
-x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead
-(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3
-t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0:
-T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1
-i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3
-(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3
-(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c
-(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3
-(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b)
-u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8))
-(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3
-t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq
-T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda
-(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1)
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6:
-T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind
-b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6))
-(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2:
-T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6:
-T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1)
-(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind
-b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x)
-(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x
-H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i
-u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0
-(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1
-H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c
-(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u
-t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u
-t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e
-H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3
-(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead
-c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head
-(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3
-(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i)
-u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2
-(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind
-b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head
-(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c:
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1:
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0:
-T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3:
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
-Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i:
-nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4:
-(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead
-(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c
-(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind
-Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat
-Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
-u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v)))
-(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_:
-T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2:
-T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v)))
-(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead
-(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl)
-x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v)
-(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0))))
-(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c
-(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
-(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0)
-x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind
-Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda
-(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c
-(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
-(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
-x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind
-Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead
-(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda
-(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w
-(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1))
-(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst
-t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u
-t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0
-H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
-(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w
-(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u
-t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0))
-(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead
-(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x
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-g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7))
-(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda
-(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e
-(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3
-H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda
-(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda
-(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl
-i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5
-(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T
-(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6:
-T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat
-Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8:
-(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2:
-T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat
-Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat
-Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x
-t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2)
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda
-(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat
-Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g
-c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0
-(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x
-t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u
-c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2
-(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3
-(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i
-u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat
-Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3))
-i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3)
-(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e
-H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e
-H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead
-(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2
-t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6)))
-(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead
-(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast)
-t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead
-(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3)))
-(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0
-(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda
-(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast)
-u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_:
-T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_:
-T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead
-(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5
-(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11:
-(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda
-(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat
-Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3
-(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i
-u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1)
-(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u
-c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1
-(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3
-t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0
-(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c
-(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat
-Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead
-(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3
-i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst
-i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2
-t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
-
-theorem ty3_csubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
-t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1
-(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g
-c2 t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2:
-C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1
-(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
-
-theorem ty3_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1
-t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e
-(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2
-t)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
-(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
-(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2
-H1) e H0))))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/defs.ma".
-
-include "LambdaDelta-1/pc3/props.ma".
-
-theorem ty3_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
-(TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
-(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t
-x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda
-(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0:
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
-(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u:
-T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1
-t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e:
-T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9
-\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in
-(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2
-H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
-(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in
-T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _)
-\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in
-(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g
-n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d
-(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_:
-(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T
-(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t))
-H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t:
-T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d
-(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5
-\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0
-(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda
-(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort
-(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let
-H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
-(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6)))))))))))))
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w
-u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n))
-u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
-Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n))
-(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v)
-(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w
-(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
-(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort
-(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort
-n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
-H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2))
-H6))))))))))) c y x H0))) H))))).
-
-theorem ty3_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
-(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
-(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
-(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t
-x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
-(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0)))))))
-(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1)
-t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e
-u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3
-c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
-T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u
-t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u
-t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
-(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6:
-(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n)
-H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift
-(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7)
-in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n)
-H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
-t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
-e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
-T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
-(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
-t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
-T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
-(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0
-(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
-(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
-t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
-e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13
-H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0)
-t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
-e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
-T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0
-(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst)
-x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
-(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0
-(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10))))))))))))))))
-(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef
-n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in
-(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
-T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda
-(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda
-(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next
-g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t:
-T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S
-n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
-T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
-(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4:
-(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 |
-(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef
-n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d
-(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C
-T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O
-t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
-t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
-(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3
-C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O
-u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O
-t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
-e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5))))))))))))
-(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3
-g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr)
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S
-n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d
-(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
-T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5
-\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
-with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _)
-\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0
-(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in
-(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda
-(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
-t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0
-(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3
-C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O
-t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
-t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
-T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0
-(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
-n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
-T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
-t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
-(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to
-(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead
-c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3
-C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind
-b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5:
-(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind
-b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda
-(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead
-(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
-t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
-T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
-n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t:
-T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
-(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u
-t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind
-Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
-T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
-(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
-t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6
-\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in
-(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
-T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u
-t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
-e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
-T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u
-t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
-e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
-n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
-(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
-T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0:
-T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
-n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
-(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
-T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
-(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T
-(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat
-Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t)
-(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
-T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2)))))
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
-t)))))) H6))))))))))) c y x H0))) H))))).
-
-theorem ty3_gen_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
-T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda
-(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_:
-T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind b) u) t1 t2))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
-T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq
-T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2
-T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0:
-(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda
-(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3:
-T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind
-b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u
-t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
-u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3:
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0:
-T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0
-(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
-c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
-t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
-t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind
-b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u
-t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind
-b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead
-(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4
-(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3
-g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T
-(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_:
-T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3
-g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
-t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3:
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0)
-t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b)
-u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead
-(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4)))
-(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1
-(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13))))))
-H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
-(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m)))))
-(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n
-c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0
-t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
-(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_:
-T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
-(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind
-b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1)
-H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead
-(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0
-u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
-t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst)
-u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0
-(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d
-(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0)))
-(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1
-t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def
-(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind
-(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2)
-(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0)))
-(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
-H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1:
-(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T
-T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t)))
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda
-(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
-(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3)
-t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4)))
-(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind
-b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind
-b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return
-(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0
-| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0
-t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
-(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0)
-u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0)
-u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10:
-(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead
-(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead
-c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5:
-T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_:
-T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in
-(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3
-t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1
-(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3
-(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda
-(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_:
-T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10)
-in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0)
-t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2))))
-(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0
-(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
-(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4)
-t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5)))
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind
-b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3:
-T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T
-u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
-(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_:
-T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0
-(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3:
-T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
-(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5)))
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))
-(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
-t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u
-t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))
-t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10))))))))
-H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0:
-T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u
-t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
-u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2:
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0
-t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
-(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0
-t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2:
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5:
-(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T
-(THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3
-c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t)))))
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0:
-C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda
-(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_:
-T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2
-t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
-(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_:
-T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g
-(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2
-t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0)
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind
-(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4)
-(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t)))
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))
-H6))))))))))) c y x H0))) H))))))).
-
-theorem ty3_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
-T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u:
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
-T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead
-(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T
-(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind
-Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind
-Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y:
-T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
-(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u
-t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u
-t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0:
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
-(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u:
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0))
-t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda
-(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat
-Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead
-(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
-T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
-T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u
-(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u
-(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq
-T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3:
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda
-(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda
-(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7)
-in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead
-(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w
-v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0:
-T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
-T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0
-(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0:
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
-t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13:
-(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0
-(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0:
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w
-(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10))))))))))))))))
-(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat
-Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w
-v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0
-(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda
-(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind
-Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u
-(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
-T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0:
-T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead
-(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
-(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O
-t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0
-t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5)))))))))))
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
-d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T
-(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind
-Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind
-Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w
-u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5
-\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in
-(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0:
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda
-(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
-(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0:
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1
-t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
-(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w
-(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g
-(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq
-T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
-(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0:
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u
-t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0
-t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6)))))))))))))
-(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0
-u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
-(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0
-t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0
-t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0:
-T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u
-t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T
-(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
-Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0:
-T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_:
-T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead
-(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _)
-\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0)
-(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 |
-(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let
-H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
-(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
-(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0:
-T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0:
-T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T
-v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let
-H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to
-(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
-(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v
-(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
-u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0
-u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0:
-T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1))
-(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda
-(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda
-(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0:
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl)
-w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v
-(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
-u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10
-H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat
-Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead
-(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t:
-T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_:
-T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
-t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda
-(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
-t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T
-(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T
-(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
-(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
-True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda
-(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
-(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v
-(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))
-H6))))))))))) c y x H0))) H)))))).
-
-theorem ty3_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall
-(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0:
-T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2))
-(\lambda (t0: T).(ty3 g c t2 t0))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T
-(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3
-T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3
-g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0:
-(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0
-(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
-(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast)
-t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t))
-(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2
-t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
-t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
-(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3
-t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal
-T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def
-(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T
-(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3
-g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2
-t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3
-(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat
-Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4
-t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))
-(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0:
-T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3
-g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4:
-T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2)
-H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m:
-nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def
-(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in
-(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort
-(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
-t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
-(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3
-T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3
-g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef
-n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0
-(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1
-t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0
-(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u
-t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda
-(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2))
-(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead
-(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead
-(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead
-(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda
-(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
-Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
-t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
-(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat
-Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead
-(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))
-(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T
-(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T
-(THead (Bind b) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
-Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda
-(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat
-Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0
-t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
-(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3
-T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u
-t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2
-t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2
-t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
-(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
-\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T
-(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w
-(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0:
-T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat
-Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g
-c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4:
-(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0
-(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
-(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0)
-(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0)
-(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
-t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9
-\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to
-(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let
-H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11
-\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to
-(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_:
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let
-H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r
-T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5
-t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda
-(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T
-t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g
-c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g
-c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat
-Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2))
-(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2))
-H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
-
-theorem tys3_gen_nil:
- \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T
-(\lambda (u0: T).(ty3 g c u u0))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil
-u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_:
-TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda
-(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq
-TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0:
-T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList
-TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda
-(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts:
-TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to
-(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t
-ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee:
-TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
-\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind
-(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
-
-theorem tys3_gen_cons:
- \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall
-(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts
-u)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda
-(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts)
-(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u)
-(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind
-g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to
-(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1:
-T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t
-ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
-return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
-\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0)
-(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1:
-(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0
-u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0)
-(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t
-ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList
-return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _)
-\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal
-TList TList (\lambda (e: TList).(match e in TList return (\lambda (_:
-TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1]))
-(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def
-(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land
-(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList
-ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind
-T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3
-g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/arity_props.ma".
-
-include "LambdaDelta-1/pc3/nf2.ma".
-
-include "LambdaDelta-1/nf2/fwd.ma".
-
-theorem ty3_gen_appl_nf2:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
-T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u:
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
-T).(nf2 c (THead (Bind Abst) u t))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
-T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda
-(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
-x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u:
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
-T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
-x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g
-c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in
-(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0
-x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl)
-w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v
-(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))
-(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda
-(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def
-(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t:
-T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead
-(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t:
-T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3
-g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t)))))
-(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c
-(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind
-Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c
-(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda
-(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
-x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
-T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c
-(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def
-(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u:
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
-T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6:
-T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c
-x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
-b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10
-(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13
-(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda
-(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u:
-T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u:
-T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c
-(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead
-(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6))
-(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w
-Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead
-(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind
-Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5
-x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2
-(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3))))))))
-(ty3_gen_appl g c w v x H))))))).
-
-theorem ty3_inv_lref_nf2_pc3:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c
-(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to
-((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda
-(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t
-u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c
-u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda
-(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2:
-T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift
-(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2
-c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T
-(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda
-(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to
-((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T
-(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0
-t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda
-(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10
-\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11
-\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to
-(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0:
-T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def
-(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y
-\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2
-H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq
-T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2:
-T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m))
-u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in
-(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5)))))))))
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g
-d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
-T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
-i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
-(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7:
-(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
-i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
-O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
-(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
-n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10
-(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))))))))))))))
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
-d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
-T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
-i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
-(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7:
-(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
-i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
-O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
-(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
-n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0
-(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y
-d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2
-(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq
-T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i)
-O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0:
-T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda
-(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S
-i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
-i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to
-(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b:
-B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
-u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u)
-t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0
-(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
-u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda
-(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0
-u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T
-(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T
-(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda
-(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
-(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2)
-\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
-u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
-(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to
-(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to
-(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq
-T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl)
-w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead
-(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead
-(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0:
-T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda
-(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
-t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0
-t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda
-(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to
-((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T
-(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T
-(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast)
-t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0
-(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2
-t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T
-u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
-
-theorem ty3_inv_lref_nf2:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c
-(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0:
-T).(eq T u (lift (S i) O u0))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1:
-(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
-
-theorem ty3_inv_appls_lref_nf2:
- \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1:
-T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to
-((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S
-i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u))
-u1))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t:
-TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t
-(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u:
-T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t
-(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H:
-(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c
-u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in
-(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u:
-T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1)))
-(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def
-(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r
-T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i)
-O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda
-(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u)
-(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2))))))))
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall
-(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef
-i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
-(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u))
-u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c
-(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t
-(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T
-T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind
-Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat
-Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_:
-T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst)
-u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u:
-T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u)))
-u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat
-Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat
-Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t
-x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def
-(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
-x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3
-c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1)))
-(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0)
-x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def
-(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c
-(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i)
-O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O
-u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
-(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O
-x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead
-(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
-(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S
-i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c
-(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t
-Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
-
-theorem ty3_inv_lref_lref_nf2:
- \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c
-(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i
-j)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda
-(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda
-(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0
-H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift
-(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S
-i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
-in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x
-(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
-(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
-j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
-(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S
-i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i)
-j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4)))))
-H2))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/arity.ma".
-
-include "LambdaDelta-1/pc3/nf2.ma".
-
-include "LambdaDelta-1/nf2/arity.ma".
-
-definition ty3_nf2_inv_abst_premise:
- C \to (T \to (T \to Prop))
-\def
- \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi:
-T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs:
-TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w
-u)) \to False)))))))).
-
-theorem ty3_nf2_inv_abst_premise_csort:
- \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise
-(CSort m) w u)))
-\def
- \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda
-(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst)
-wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs
-(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d
-(Bind Abst) wi) H False))))))))).
-
-theorem ty3_nf2_inv_all:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
-u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
-t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H)
-in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
-(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda
-(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w)
-u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2:
-(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g
-c t x H2 H0)))) H1)))))))).
-
-theorem ty3_nf2_inv_sort:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t
-(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort
-n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda
-(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def
-(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2
-T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u:
-T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))
-(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
-m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
-t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
-i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w)
-u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind
-Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w:
-T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda
-(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0
-x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0)
-x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H
-(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1)
-(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
-(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2:
-T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_:
-T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead
-(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n))))
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
-Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7:
-(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0
-x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst
-c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat
-(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n:
-nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
-(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c
-x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n:
-nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n)))
-(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
-m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
-t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
-i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def
-(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in
-(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T
-t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x)
-(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort
-n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda
-(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda
-(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x)
-(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
-(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
-(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
-i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda
-(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x))
-(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g
-c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda
-(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda
-(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n:
-nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
-TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef
-x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6
-\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat
-Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1))
-(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
-(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda
-(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n:
-nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
-(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda
-(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
-x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))
-x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3)))))))
-H2)) H1)))))))).
-
-theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux:
- \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise
-c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead
-(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to
-(ty3_nf2_inv_abst_premise c w2 u2))))))))
-\def
- \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d:
-C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi))
-\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi))
-(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2:
-T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind
-Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi:
-T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst)
-wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift
-(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t
-(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t
-(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl)
-vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst)
-w1 u1) H0))))))))))))))).
-
-theorem ty3_nf2_inv_abst:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u:
-T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to
-((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_:
-T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g
-c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v
-u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w)
-v))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u:
-T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c
-t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w
-u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in
-(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq
-T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c
-w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))))
-(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v:
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
-(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0:
-T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2
-c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0)
-u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind
-Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0:
-T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda
-(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
-(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t
-(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2
-(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3
-g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r
-T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v:
-T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
-(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst)
-w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst)
-x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
-(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T
-(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x)))
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v:
-T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v))))
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
-T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3:
-T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w
-x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T
-(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead
-(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0)))
-(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2)))
-(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
-(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
-(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
-(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4)
-(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g
-(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b:
-B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda
-(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w
-v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c
-x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind
-b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def
-(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w
-H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst)
-t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v:
-T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v))))
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
-T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v:
-T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v))))
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
-T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w
-x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w))
-H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c
-x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9))))
-(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t
-H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort
-n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda
-(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
-(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let
-H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H
-(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda
-(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
-(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c
-(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq
-T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
-g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v
-u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t
-H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
-nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda
-(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
-TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
-nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t
-(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
-(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
-(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0
-(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef
-x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind
-Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads
-(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v:
-T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
-(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u
-(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind
-Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda
-(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind
-Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w
-(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1))
-(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T
-T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1))
-(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v))))))))
-(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c
-(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to
-((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
-(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v))))
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
-T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2:
-T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda
-(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
-Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
-T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
-(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0)))))
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2
-T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x
-v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C
-T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O
-t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
-(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
-Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
-T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef
-x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda
-(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O
-x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind
-Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7
-(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
-Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda
-(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c
-(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e:
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0)
-(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
-T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
-T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda
-(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
-(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
-(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
-Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda
-(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15:
-(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let
-H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind
-(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
-Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13))
-(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0:
-T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2:
-T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to
-((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
-(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v))))
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
-T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2:
-T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
-(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13:
-(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda
-(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind
-Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl)
-t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_:
-T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead
-(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v))))
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
-T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4:
-T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))
-(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1
-(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let
-H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let
-H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T
-(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1))
-(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4)))
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T
-(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
-Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
-(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
-(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
-Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads
-(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c
-x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_:
-(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T
-(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T
-(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
-Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
-(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
-(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
-Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1))
-(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v:
-T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil
-(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
-g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
-x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))
-H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads
-(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda
-(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3
-(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
-g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
-x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
-v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1))
-(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2
-(THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda
-(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons
-t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0:
-T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind
-Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
-v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat
-Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t
-H5))))))) H4)) H3))))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/csubt/ty3.ma".
-
-include "LambdaDelta-1/ty3/subst1.ma".
-
-include "LambdaDelta-1/ty3/fsubst0.ma".
-
-include "LambdaDelta-1/pc3/pc1.ma".
-
-include "LambdaDelta-1/pc3/wcpr0.ma".
-
-include "LambdaDelta-1/pc1/props.ma".
-
-theorem ty3_sred_wcpr0_pr0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
-t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2)
-\to (ty3 g c2 t2 t)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
-(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
-(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to
-(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0:
-T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c
-c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u:
-T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2:
-C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4
-t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0
-c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2
-H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2
-H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2:
-C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m)
-t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m))))
-(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda
-(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
-Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2:
-((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g
-c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2:
-T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3:
-T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda
-(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2
-(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5:
-(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda
-(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)))))))
-(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n
-H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0:
-T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2)
-\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2:
-C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n)
-t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u)))
-(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind
-Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind
-Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g
-c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u
-(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S
-n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n)
-O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1
-H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n
-H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda
-(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to
-(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b:
-B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b)
-u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2)
-\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2:
-C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead
-(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5:
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u
-t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with
-[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u
-t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda
-(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8:
-(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda
-(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u
-(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u
-(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead
-(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow
-(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq
-T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
-\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead
-(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead
-(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in
-T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind
-b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T
-t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to
-(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind
-T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to
-((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))
-(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead
-(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead
-(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind
-T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to
-(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda
-(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3
-t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x:
-T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7:
-T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6)
-(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind
-b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x)
-(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2
-t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3
-(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17))
-(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u
-u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b)
-u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b))
-t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b)
-u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4
-H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind
-b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7)
-\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0
-t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6)
-t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0
-t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2)
-H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to
-((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) |
-(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq
-T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u
-t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
-b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2)
-H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2)
-\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7
-H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T
-(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead
-(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
-\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5)
-(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind
-Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B
-(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
-(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u
-t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to
-((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to
-((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda
-(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T
-(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O
-u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq
-T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4)
-\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4
-(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w)
-t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to
-((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u
-t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda
-(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0:
-B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7:
-T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def
-(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr
-H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7))
-(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x:
-T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda
-(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr)
-u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead
-c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead
-(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22)
-(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1
-c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20
-(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18)
-c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2
-t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind
-Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2
-(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18)))))
-(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u)
-(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4
-H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11))
-H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq
-T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda
-(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
-((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8)
-\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in
-lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
-t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
-t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7)
-\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u
-t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0
-| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5))
-(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 |
-(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in
-(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2)
-\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4
-(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda
-(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b
-Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda
-(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_:
-T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4
-(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda
-(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind
-b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5
-t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0
-(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8
-t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda
-(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in
-(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind
-b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead
-c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4)
-t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B
-Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda
-(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def
-(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O
-(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2
-(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4)
-(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S
-O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4
-(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25:
-(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u
-t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind
-T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O
-(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat
-(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S
-O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3)
-(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3
-x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead
-(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2
-(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2
-(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1
-u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2
-(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u
-(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u))))))))))))
-H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2
-(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S
-O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return
-(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in
-H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead
-c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift
-(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u)
-(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind
-Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
-(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O
-x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0
-x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void)
-u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1)
-(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def
-(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S
-O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead
-(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S
-O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O
-x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2
-(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl
-x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u)
-(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5
-t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3
-(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b))
-(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4
-H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10))
-H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead
-(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let
-H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0
-t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6
-(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4)))))))))))))))))
-(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w
-u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
-w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda
-(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2:
-C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead
-(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
-c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6
-\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_:
-(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g
-c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl
-t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda
-(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T
-t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))
-(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl)
-w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u
-t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v
-(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) |
-(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1
-t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let
-H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5)
-\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _)
-\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K
-(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead
-k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
-Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind
-T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2)
-\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
-(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v
-(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to
-((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u
-t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T
-(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to
-(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda
-(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3
-g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3
-g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda
-(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6:
-T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind
-Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w
-(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
-(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda
-(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat
-Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u
-x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0
-(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead
-(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2
-H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead
-(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1
-c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0)))))))))
-(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst)
-u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1
-(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6
-H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T
-(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w
-v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow
-(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat
-Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _)
-\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead
-(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind
-Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to
-((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u
-t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T
-(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4)
-t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
-(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4)
-t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to
-((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u
-t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def
-(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall
-(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3
-(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5:
-T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12)
-in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g
-c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
-(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0)
-x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind
-Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda
-(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2
-(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u
-x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind
-Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2
-(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_:
-T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3
-g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
-(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u
-t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind
-Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1:
-T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
-(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0
-u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b)
-u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
-(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
-(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0
-H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u
-(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3
-H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead
-(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind
-Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0))
-(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind
-Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
-(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0))))))))
-(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead
-(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0
-(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x
-H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0)
-(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0
-t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) |
-(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq
-T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w
-v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3)
-| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow
-t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v)
-H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
-| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1
-t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T
-(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to
-((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead
-(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3)
-v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to
-((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
-Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b
-Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5
-(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq
-B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda
-(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3:
-C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead
-(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def
-(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2
-(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead
-(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x:
-T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in
-(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u
-t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5:
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w
-(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
-(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda
-(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5:
-T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0))))
-(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda
-(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind
-Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead
-(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2
-x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def
-(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3
-(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27)
-(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u
-t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u
-t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x)))
-(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O)
-(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S
-O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T
-T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind
-Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6:
-T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5:
-T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S
-O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u
-t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b)
-u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33:
-(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g
-(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O)
-t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
-(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
-(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0
-H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S
-O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat
-Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind
-Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b)
-u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17)
-(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2)
-u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind
-Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O)
-O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst
-(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead
-(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0))))
-(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O)
-O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0))
-(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead
-(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl
-v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O
-(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0))))
-(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w
-H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind
-Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u
-t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0)
-(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat
-Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0))
-(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b)
-u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31)))))))))
-(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind
-b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g
-Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind
-Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19
-(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7
-H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq
-T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T
-(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1
-t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v)
-H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to
-((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w
-(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4
-H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3))
-(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind
-T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to
-((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
-(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0)
-\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl)
-w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast)
-u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
-F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
-(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g
-c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in
-(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2))))))))))))))))
-(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2
-t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0
-t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3
-t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0
-t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
-c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let
-H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda
-(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to
-(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow
-(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5
-t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to
-(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat
-Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g
-c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl
-t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat
-Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda
-(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T
-(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _)
-\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead
-(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead
-(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat
-Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to
-((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5
-t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1
-t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat
-Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead
-(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda
-(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7
-t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T
-(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda
-(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0
-t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T
-(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead
-(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv
-g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0
-(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat
-Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6
-t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2
-H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3)
-(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2)
-(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2
-t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1
-(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6
-H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead
-(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda
-(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead
-(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
-\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T
-(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2
-t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2
-H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1
-(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
-\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
-True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in
-(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0
-t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8
-H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T
-(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T
-(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1
-t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3
-t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2)
-\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast)
-t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow
-(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast)
-t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b)
-u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to
-((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7)))
-| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u
-t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7)
-\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in
-((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7
-_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7)
-in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5
-t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5
-t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g
-c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4
-(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))
-(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3
-g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2
-t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0)
-(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4
-H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat
-Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0
-(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2
-H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead
-(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
-
-theorem ty3_sred_pr0:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall
-(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g:
-G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1
-t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
-
-theorem ty3_sred_pr1:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall
-(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3:
-T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g:
-G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))))
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5:
-T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c:
-C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g:
-G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t
-(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
-
-theorem ty3_sred_pr2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
-(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g:
-G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g:
-G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3
-t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
-G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0
-(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t
-H2)))))))))))))) c t1 t2 H)))).
-
-theorem ty3_sred_pr3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
-(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall
-(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g:
-G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
-(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c
-t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3:
-(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2
-H)))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/pr3.ma".
-
-theorem ty3_cred_pr2:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1
-v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
-(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
-(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind
-b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0:
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b:
-B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b)
-t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0
-(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0
-(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
-u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda
-(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0
-t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead
-c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl
-c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead
-c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0)
-(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1
-v2 H))))).
-
-theorem ty3_cred_pr3:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1
-v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
-(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
-(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b:
-B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to
-(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b:
-B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b)
-t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1
-t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b:
-B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to
-(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0:
-T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b
-t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
-
-theorem ty3_gen_lift:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h:
-nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop
-h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2:
-T).(ty3 g e t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T
-(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e:
-C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x))
-(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c
-y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall
-(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x))
-(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall
-(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to
-(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t
-t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall
-(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
-t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda
-(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e:
-C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t))
-(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3:
-T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall
-(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
-(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e
-x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7:
-(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2:
-T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h
-x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda
-(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def
-(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let
-H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda
-(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0
-t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda
-(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4)
-t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2
-H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
-nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift
-h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort
-m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort
-(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e
-(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
-T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1
-(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0
-(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0:
-C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind
-Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3:
-((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall
-(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2)
-t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
-(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
-\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
-h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
-(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda
-(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
-(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0
-t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5
-(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
-(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
-T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
-(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
-(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
-\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
-t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
-(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
-t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
-(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
-(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n))
-x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n))
-t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0
-(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))
-(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4)
-t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S
-n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift
-(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
-(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t)))
-(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift
-(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n)
-O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus
-x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4))
-(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g
-n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16)))))))))
-(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9)))
-H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n
-h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
-T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0
-(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
-T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
-(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift
-(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda
-(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h)))
-(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n
-(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0
-(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus
-x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n
-h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O
-x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n
-H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e
-d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2))
-x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
-(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst)
-u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall
-(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e:
-C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t))
-(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda
-(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6
-\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
-h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
-(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda
-(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
-(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0
-t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5
-(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v))))
-(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
-T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11:
-(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3
-(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14
-\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T
-t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T
-(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4
-t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
-(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
-(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda
-(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e
-(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h
-(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h
-(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda
-(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2))))
-(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3
-d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2
-x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T
-(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S
-n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
-(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift
-h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g
-e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus
-x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus
-(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3
-c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus
-n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2)))
-(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S
-n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O
-(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1
-(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst
-c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
-(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h)
-n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
-t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le
-(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T
-(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h
-x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
-T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T
-(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O
-u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0
-O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
-O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h))
-(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h
-(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h))
-O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus
-O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h)))))
-(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0
-(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6))))))))))))))))
-(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u
-t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
-c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b:
-B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b)
-u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift
-h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T
-(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4:
-T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5:
-(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6:
-(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
-(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1
-y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4:
-T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0
-(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9:
-(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda
-(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u
-t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda
-(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to
-(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda
-(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3
-g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2
-(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3)
-H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b)
-t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T
-u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1)
-x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0)
-e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4)
-t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let
-H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
-nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
-(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0
-x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda
-(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2)
-(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind
-b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16
-\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda
-(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3)))
-(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4:
-T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2
-x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e
-(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4:
-T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda
-(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0
-(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e
-(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0
-(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead
-e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b)
-x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
-(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))
-(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2
-T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2)
-t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b)
-x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5))
-(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2
-c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind
-b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21))))
-(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u
-H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5)))))))))))))))))
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w
-u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
-c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v:
-T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u
-t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3
-c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0
-t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead
-(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0
-e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
-Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0))))
-(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t))))
-(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
-(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1
-x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl)
-x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead
-(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2))))
-(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5:
-nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t)))
-(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11
-\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3
-(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall
-(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0:
-C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u))
-(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13
-\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in
-(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
-(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2:
-T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1
-(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0
-(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
-(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16:
-(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6)
-in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u
-t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift
-h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t))))
-(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5:
-T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda
-(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3
-e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u
-(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda
-(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
-(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst)
-x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall
-(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1)
-x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2:
-T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
-(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3
-e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2:
-T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_:
-T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g
-(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
-t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda
-(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda
-(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25:
-(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7
-x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl)
-(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead
-(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))
-(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst)
-x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead
-(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7))
-(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h
-(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t)))
-(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind
-Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6))
-(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind
-Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1
-(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead
-(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4
-H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6)
-(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind
-Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9
-H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20)))))))))
-(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19)))))))
-(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0
-H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0:
-C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda
-(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to
-(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h
-x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0:
-T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall
-(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
-(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e
-x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T
-(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6:
-(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
-(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h
-x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda
-(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq
-T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1
-x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast)
-x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead
-(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind
-T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5
-x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3
-c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h
-x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3
-(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall
-(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0:
-C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t))
-(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13
-\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in
-(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0
-(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead
-(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g
-c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2
-(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4))
-\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0
-(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12
-(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3))
-e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2)))
-(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1
-t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead
-(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1
-x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1
-(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0
-(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4:
-T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda
-(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda
-(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2
-T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1
-x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat
-Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2))
-(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1
-c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat
-Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e
-x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21)))))
-H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1
-H5))))))))))))))) c y x H0))))) H))))))).
-
-theorem ty3_tred:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
-t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
-(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T
-(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1:
-(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g
-c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))).
-
-theorem ty3_sconv_pc3:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
-u2) \to (pc3 c t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
-u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
-(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x:
-T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def
-(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g
-t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))).
-
-theorem ty3_sred_back:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c
-t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2
-t) \to (ty3 g c t1 t)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda
-(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
-(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t
-t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g
-c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t
-H1)))) (ty3_correct g c t2 t H1)))))))))).
-
-theorem ty3_sconv:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
-u2) \to (ty3 g c u1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
-u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
-(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda
-(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back
-g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/fwd.ma".
-
-include "LambdaDelta-1/pc3/fwd.ma".
-
-theorem ty3_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e
-t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c
-e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
-(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0:
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h
-d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3
-g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h:
-nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
-t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d:
-nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h
-d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5)
-(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m:
-nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop
-h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort
-(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0
-(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort
-(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n:
-nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
-(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
-t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h:
-nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0
-t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3:
-(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0
-(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le
-n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0)
-in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0)))
-(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_:
-C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0
-(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1:
-C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0
-x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind
-nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S
-n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0
-(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1
-(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0
-(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t)))
-(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
-d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T
-(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t))))
-(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef
-n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0
-(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda
-(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t))))
-(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x
-(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n))
-t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
-(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
-n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
-(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
-H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
-h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat
-(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
-t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef
-(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
-(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u
-(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus
-h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n)
-h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
-n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
-(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
-d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda
-(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall
-(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u)
-(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h:
-nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0
-(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5
-\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3
-(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_:
-C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n)
-e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst)
-u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda
-(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop
-h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let
-H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S
-(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0
-h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0
-(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h
-(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S
-n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
-h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x
-d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S
-n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3
-g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift
-h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat
-d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S
-n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0
-(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus
-d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
-(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
-n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
-(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
-H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
-h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat
-(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
-u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef
-(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
-(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u
-(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus
-h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n)
-h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
-n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
-(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
-d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
-t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3
-g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0
-(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d:
-nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
-(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0
-t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d
-u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b)
-(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u)
-(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead
-c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h
-d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead
-(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c:
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1:
-((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
-(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall
-(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0
-(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0:
-C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0
-c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v))
-(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind
-Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat
-Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat
-Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead
-(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat
-Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w)
-(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl
-g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t)
-(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift
-h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t))
-(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t))
-(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
-w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t)
-h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h
-d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda
-(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d
-t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3:
-((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
-(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d:
-nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead
-(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3
-g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast)
-(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead
-(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0
-(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat
-Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4
-t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0))
-(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))).
-
-theorem ty3_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
-t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda
-(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0:
-C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda
-(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3:
-T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g
-c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g
-c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T
-(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m)))
-(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda
-(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3
-g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda
-(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0
-(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n)
-(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind
-Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T
-(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0
-(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n)
-(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u:
-T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda
-(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T
-(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in
-(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T
-(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x:
-T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda
-(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g
-c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w:
-T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda
-(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
-c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0
-(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0:
-T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w
-(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u
-x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind
-Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead
-(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind
-Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0
-(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u
-t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t
-t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind
-Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0
-(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10:
-(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0
-(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w
-(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1
-(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0
-H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g
-c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3:
-(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T
-(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat
-Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T
-(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x
-t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))).
-
-theorem ty3_unique:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
-t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
-(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0:
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
-(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0:
-T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall
-(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0
-t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s
-c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m:
-nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g
-c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
-(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t:
-T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0
-t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n)
-t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
-T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1)))))
-(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0
-(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
-T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
-(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
-t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g
-e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda
-(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1
-x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n
-c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n
-H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind
-Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr)
-x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0)
-\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
-(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
-T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def
-(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
-(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d
-H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d
-H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O
-(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9)))))))))
-H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
-T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst)
-u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))
-(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0
-(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def
-(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead
-x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0
-(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst)
-x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1)
-H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4))
-(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0:
-C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind
-Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_:
-((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2:
-T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))
-(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift
-(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
-(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4:
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
-(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
-c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
-(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
-T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
-(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O
-u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3
-c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr)
-x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind
-Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
-(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in
-C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
-(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0
-(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda
-(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2))))
-(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1
-t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
-T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O
-x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7:
-(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
-(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal
-C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0)
-(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
-x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match
-e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
-t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
-(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0:
-T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def
-(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def
-(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in
-(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind
-Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3
-g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2
-n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda
-(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to
-(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3:
-T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2
-t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0)
-t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
-u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda
-(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead
-(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0
-(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3
-g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead
-(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5))))))
-(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w:
-T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2:
-T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3:
-((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t)
-t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v)
-t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0:
-T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_:
-T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t))
-t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat
-Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead
-(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl)
-w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0
-t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3
-(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2
-H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
-(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to
-(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda
-(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4:
-T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda
-(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0
-t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4)
-(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2)
-t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t
-(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0
-(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4
-H4)))))))))))) c u t1 H))))).
-
-theorem ty3_gen_abst_abst:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
-(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2
-T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst)
-u) t1 t2))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u
-t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T
-(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u)
-t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2)
-x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst)
-u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3:
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda
-(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
-x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind
-Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c
-(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda
-(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind
-Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3
-g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
-(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5:
-(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y
-\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g
-c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5
-(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y))))))))
-(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g
-Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind
-Abst) u t2) H))))))).
-
-theorem ty3_typecheck:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t
-v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H:
-(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u:
-T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g
-c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))
-(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v
-H)))))).
-
-theorem ty3_getl_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
-u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t
-t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d
-(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
-T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2)
-\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d
-(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda
-(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
-t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0
-t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
-nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3:
-T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b:
-B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b)
-v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
-nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0:
-(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v:
-T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m
-H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
-c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0
-t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
-T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
-w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
-(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
-(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n
-i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w)))
-(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7
-\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v)))
-H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1:
-C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
-Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
-(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
-(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
-(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
-(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2)
-\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono
-c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12:
-(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v
-(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T
-u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def
-(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d
-H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w))))
-(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0)
-u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0
-H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n
-H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
-u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall
-(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall
-(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to
-(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda
-(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda
-(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0
-(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T
-(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq
-T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0:
-nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C
-(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind
-b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7))
-in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
-(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
-Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B
-(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])]))
-(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
-Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0)
-(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0
-(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d
-d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind
-b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w:
-T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n
-c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T
-(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0:
-B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda
-(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9))))))
-(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda
-(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1:
-((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to
-(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
-v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b:
-B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
-u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
-nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v:
-T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda
-(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda
-(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0:
-B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind
-b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
-(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3
-(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
-T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w:
-T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead
-(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda
-(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
-u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T
-t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0
-d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind
-b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T
-(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
-(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
-T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s
-(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0
-(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
-T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
-T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
-x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
-v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0
-u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
-(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0:
-T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b:
-B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex
-T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0:
-T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3:
-((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to
-(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind
-b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0:
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
-Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda
-(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq
-T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T
-(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
-(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
-T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
-v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v
-t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda
-(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v)))
-(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0)))
-(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8:
-(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T
-(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
-(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead
-(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex
-T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1
-(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v
-x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0:
-T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1
-(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9:
-(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1
-H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4)))))))))))))))))))
-(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1
-t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i:
-nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3:
-((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to
-(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
-v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0:
-T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
-Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda
-(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T
-t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T
-(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4:
-T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat
-Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6:
-(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2:
-T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat
-Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w:
-T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x
-t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6))
-(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4)))
-(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda
-(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s
-(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
-T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s
-(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6))
-(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T
-(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2
-x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v
-H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4))))))))))))))))))
-c t u H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/pr3_props.ma".
-
-include "LambdaDelta-1/sty0/fwd.ma".
-
-theorem ty3_sty0:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
-t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
-(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_:
-T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0:
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
-(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda
-(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3:
-((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_:
-(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0
-H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda
-(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in
-(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in
-(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t))
-(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
-(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
-u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall
-(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda
-(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in
-(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C
-T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
-(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g
-e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift
-(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
-u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O
-t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2)
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
-(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8:
-(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
-(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal
-C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
-(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
-x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match
-e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _
-t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
-(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0:
-T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def
-(eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 u0 H12) in (let H16
-\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0)))
-H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(sty0 g c1 u0
-x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2
-H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O
-u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2)
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0
-(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq
-T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2
-(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind
-C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match
-b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
-\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
-False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0)
-n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift
-(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
-c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0
-t)).(\lambda (_: ((\forall (t2: T).((sty0 g d u0 t2) \to (ty3 g d u0
-t2))))).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 (TLRef n) t2)).(let H_x
-\def (sty0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T
-(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1
-t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
-O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2)
-(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_:
-T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T
-(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
-Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1
-t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n)
-O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_:
-(sty0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def
-(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift
-(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C
-(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind
-Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr)
-x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
-[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
-(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0
-(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind
-(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda
-(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0
-(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0:
-T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T
-t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e:
-C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0
-(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda
-(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1
-x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T
-(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1)
-(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d
-(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
-(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
-C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead
-d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind
-Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0)
-(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
-x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def
-(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0
-H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7
-u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O
-t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1
-(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1:
-C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1
-H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda
-(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall
-(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda
-(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2
-t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4)
-\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4:
-(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0
-t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead
-c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4)))
-(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g
-(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0
-x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x)
-H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead
-(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8)))))
-H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda
-(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to
-(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
-(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2)
-\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead
-(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5
-\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3:
-T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2)
-(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead
-(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead
-(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0:
-T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9
-\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T
-(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
-(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T
-(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead
-(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T
-(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead
-(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12:
-(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3:
-T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_:
-T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3
-g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead
-(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0
-(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda
-(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat
-Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w
-u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind
-Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl)
-w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v
-t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead
-(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2
-H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0
-w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5))))))))))))))
-(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2
-t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2
-t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3:
-((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4:
-T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def
-(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T
-(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda
-(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead
-(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0
-t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def
-(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r
-T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3
-t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10
-\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0
-x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1))
-(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t:
-T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0
-x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead
-(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g
-c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat
-Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1
-(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y))))
-(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/props.ma".
-
-include "LambdaDelta-1/pc3/subst1.ma".
-
-include "LambdaDelta-1/getl/getl.ma".
-
-theorem ty3_gen_cabbr:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
-t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
-(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to
-(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2))))))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
-e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
-C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u:
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0:
-C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T
-T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u:
-T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e:
-C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0))
-\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d
-a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S
-O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0:
-T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr)
-u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a:
-C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7)
-in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O)
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda
-(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d
-u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e
-u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d
-x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a
-x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u
-(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9
-H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0
-H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0:
-C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
-nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
-C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
-d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort
-m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort
-(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
-y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t:
-T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort
-m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
-T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m)))
-(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g
-a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
-(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t:
-T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0:
-T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0:
-C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e:
-C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
-(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
-a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
-O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0:
-nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)))
-(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0
-(le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6))
-in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr)
-u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1
-(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let
-H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d
-(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11
-\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in
-(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind
-Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u
-u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2)))
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
-O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind
-Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14:
-(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1:
-C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind
-nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S
-n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
-C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18:
-(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S
-n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S
-n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2
-e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u
-(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift
-(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1
-y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S
-n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S
-n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0
-(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4
-x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2
-(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r
-nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S
-n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O)
-n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
-u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5)
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0))
-(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O)
-d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda
-(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0))
-(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n)
-H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n)))
-(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus
-d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0
-H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt
-Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11))))))
-(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda
-(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S
-O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0:
-nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0
-(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind
-nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
-n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d
-(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0)
-(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
-(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
-(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind
-Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T
-(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u)
-(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e
-(Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T
-u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let
-H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in
-(eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda
-(c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T
-(\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift
-n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T
-(lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0))
-(subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n
-(plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0:
-T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift
-(S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n)))
-(ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge
-n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12)))))
-H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n
-(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S
-O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
-d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O))
-(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
-d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift
-(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O
-t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
-(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O))
-(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O)))
-t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0
-(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus
-d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0
-u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0
-(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0)
-(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0:
-nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S
-O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0
-n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a
-(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
-(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
-n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
-O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
-O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
-H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
-u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
-C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0))
-\to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O)
-d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift
-(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda
-(H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4:
-(csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0
-a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0
-u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
-d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
-T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat
-(minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e
-(Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d
-(Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n)))
-(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0
-(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift
-(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
-(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u)
-x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n)
-(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0
-(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst)
-d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda
-(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_:
-C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2:
-C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
-C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1
-(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d
-x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1
-(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop
-(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in
-(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0
-(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst)
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0)))
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
-O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u)
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus
-d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda
-(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0
-(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0
-(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind
-Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u
-(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
-(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
-x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n))
-x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda
-(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S
-n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0:
-nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S
-n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
-u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
-T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O)
-(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
-a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0:
-T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0
-(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S
-O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S
-n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n))
-x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O)
-(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n)
-(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5
-H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0
-H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus
-d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead
-d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r
-nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def
-(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let
-H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr)
-u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C
-(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind
-Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0)
-H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
-[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
-(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0
-(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S
-O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u)
-(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n
-(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S
-O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
-d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O))
-(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1
-d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift
-(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O
-u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
-(TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O))
-(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O)))
-t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0
-(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus
-d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0
-u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0
-(lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0)
-(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0:
-nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S
-O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0
-n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a
-(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
-(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
-n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
-O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
-O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
-H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
-T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0:
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
-C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b:
-B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
-u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d:
-nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall
-(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop
-(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3
-(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
-(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5:
-(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let
-H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead
-(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1
-d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8:
-(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d
-x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head
-(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d
-x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b)
-x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
-O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
-t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S
-O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d)
-x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T
-(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
-O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
-t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
-(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b)
-(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead
-(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3
-(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b
-x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S
-d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head
-u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S
-O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1
-H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0:
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1:
-((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
-(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a:
-C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g
-c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
-C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
-(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5:
-(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let
-H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
-t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d
-x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d
-x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6)
-in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O)
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead
-(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w
-(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d
-x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind
-Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead
-(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S
-O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u
-x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S
-O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y:
-T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
-t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst)
-x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5
-(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1
-(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4
-(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25
-\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6
-x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead
-(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in
-(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat
-Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0
-(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead
-(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl)
-(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead
-(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v
-(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat
-Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift
-(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead
-(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d
-x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind
-Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S
-d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t)
-t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t
-(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7))
-(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead
-(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d))
-(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S
-O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda
-(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e
-(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
-C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4
-t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl
-d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to
-(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
-nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
-C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S
-O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda
-(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda
-(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda
-(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let
-H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
-(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead
-(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d
-u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def
-(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift
-t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
-T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast)
-x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d
-x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3)
-t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2)
-H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d))
-(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda
-(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S
-O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat
-Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1
-H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
-
-theorem ty3_gen_cvoid:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
-t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
-(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T
-T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2))))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
-e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3:
-T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e:
-C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to
-(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u
-t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl
-d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4
-t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d
-c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0
-a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1:
-T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
-y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9:
-(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def
-(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in
-(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d
-x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S
-O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0)
-(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0
-d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15:
-(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d
-x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0:
-T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3
-(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15)
-in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0))
-H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0:
-T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
-(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift
-(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1
-H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u
-H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda
-(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e
-(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0
-a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m))
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
-(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T
-(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m
-(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g
-m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m)))
-(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n:
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n
-c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
-t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl
-d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0:
-T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void)
-u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt
-n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0
-(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e
-(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n)
-d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind
-nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S
-n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
-T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
-(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
-(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
-\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
-(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
-(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
-(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
-(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
-T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def
-(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0)
-u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
-(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n))
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n))
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0)
-(lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus
-d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t
-(\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S
-O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3)
-(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
-t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16
-x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O)
-(plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0:
-nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O)
-d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O
-x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n)
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0
-(lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
-T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n)
-(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0
-(TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift
-(S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n)))
-(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3))
-(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t
-H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0
-(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0
-(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r
-nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in
-(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
-T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
-T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
-(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0
-(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d
-(Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda
-(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0:
-nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O)
-d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O)
-d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat
-(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S
-O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq
-T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
-(lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
-(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t)
-(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T
-(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n
-(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus
-n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O
-t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O
-t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O
-n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S
-O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t)))
-(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr)
-u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le
-n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n
-(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym
-(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus
-O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
-H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
-u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
-C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0))
-\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0:
-nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a:
-C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda
-(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6
-\def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind
-Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0)
-c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S
-(minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0
-(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n))))
-(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
-C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
-T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
-(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
-(Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
-\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall
-(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop
-(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift
-(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift
-(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0:
-T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T
-(lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus
-d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S
-n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
-(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda
-(_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0))
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0
-(S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S
-O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def
-(eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0))
-H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2
-(\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S
-n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n)
-O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
-(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
-(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
-T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S
-n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
-(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0
-H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0
-H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift
-(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S
-n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8))))))))
-(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda
-(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S
-O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0:
-nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n
-(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
-n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
-u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl
-n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n
-H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind
-Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
-(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind
-Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0)
-H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
-n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
-u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
-H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S
-O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
-(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift
-(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
-y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus
-(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S
-O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda
-(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef
-(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O))))
-(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T
-(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0))
-(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n
-(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0)))
-(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n
-(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge
-n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0)
-(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1)
-n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n
-(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O))))
-(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n
-(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0:
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda
-(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
-(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda
-(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3:
-((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind
-b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0
-(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e:
-C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind
-Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def
-(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
-u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S
-O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda
-(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
-(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in
-(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1)))
-H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0:
-T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0
-(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0
-(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1
-y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda
-(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in
-(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda
-(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0
-(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0))
-(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T
-T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda
-(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda
-(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S
-O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S
-O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
-y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S
-O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17:
-(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3)
-(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead
-(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O)
-(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
-(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
-x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind
-b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d)
-x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
-(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead
-(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2))
-(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3))
-(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3
-(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u
-H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
-(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall
-(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall
-(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v
-(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0:
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a:
-C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
-v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind
-Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda
-(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5:
-(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T
-(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
-Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
-y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O)
-d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d
-x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0:
-T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in
-(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
-Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
-y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead
-(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
-x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
-Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u
-(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14
-\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2
-x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T
-(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
-x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
-Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda
-(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0
-(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in
-(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
-(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
-(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4:
-T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18:
-(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4
-x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O)
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead
-(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r
-T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18))
-in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d
-x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d
-(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind
-Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
-a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst)
-x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
-(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
-T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4
-(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4
-(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4
-x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2
-x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d
-x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind
-Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d
-x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2)
-(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u
-H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7)))))))
-H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall
-(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall
-(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
-y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3:
-((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind
-Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda
-(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
-(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda
-(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a:
-C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1:
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
-(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8:
-(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
-(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in
-(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def
-(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O)
-d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0:
-C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void)
-u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1:
-T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
-g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4
-(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T
-(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
-T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_:
-T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O)
-d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def
-(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
-t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d
-x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S
-O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
-(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda
-(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2
-x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d
-x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda
-(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast)
-(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
-T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O)
-d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def
-(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d
-H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t:
-T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1))))
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1)
-(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2:
-T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0))
-(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O)
-d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
-a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S
-O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
-(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2))))
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2)
-(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0
-x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a
-x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0))
-(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S
-O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0
-H8))))))) H6)))))))))))))))) c t1 t2 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/pr0/defs.ma".
-
-include "LambdaDelta-1/C/defs.ma".
-
-inductive wcpr0: C \to (C \to Prop) \def
-| wcpr0_refl: \forall (c: C).(wcpr0 c c)
-| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall
-(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k
-u1) (CHead c2 k u2)))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/wcpr0/defs.ma".
-
-theorem wcpr0_gen_sort:
- \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort
-n))))
-\def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n)
-x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C
-x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c:
-C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c:
-C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
-C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0))
-(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda
-(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
-c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
-(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C
-(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
-with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
-(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1))
-H5))))))))))) y x H0))) H))).
-
-theorem wcpr0_gen_head:
- \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
-(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
-C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
-\def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
-(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c:
-C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2:
-C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda
-(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0:
-C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2:
-C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))
-(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal
-C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1)
-(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
-T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1
-c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C
-(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
-T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))
-(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2:
-C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to
-(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead
-c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2:
-T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0
-k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
-\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
-(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K
-k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C
-(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3:
-T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
-T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10
-\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1
-(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda
-(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
-(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k
-u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C
-c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def
-(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1
-(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda
-(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
-(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T
-(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3))))
-(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq
-C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0
-c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C
-(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x
-H0))) H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/wcpr0/defs.ma".
-
-include "LambdaDelta-1/getl/props.ma".
-
-theorem wcpr0_drop:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
-e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
-(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
-(_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
-e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
-c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1)
-(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _)
-\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k
-u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0)
-(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1)
-(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in
-(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda
-(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k
-u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t:
-T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2)
-(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_:
-C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7))))
-H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0
-u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead
-e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
-(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b:
-B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2
-k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead
-e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1
-k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
-O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0
-x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop
-(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f:
-F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2
-k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead
-e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead
-e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
-(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0
-u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
-(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1
-(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k)
-h)))))))))) c1 c2 H))).
-
-theorem wcpr0_drop_back:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
-e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
-(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
-(_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0
-u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
-e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
-u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
-c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2)
-(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _)
-\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k
-u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0)
-(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2)
-(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in
-(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda
-(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k
-u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t:
-T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1)
-(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_:
-C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7))))
-H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0
-u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead
-e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
-(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b:
-B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2
-k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead
-e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1
-k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
-O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1
-u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop
-(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f:
-F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2
-k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead
-e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead
-e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
-(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0
-x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
-(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1
-(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k)
-h)))))))))) c2 c1 H))).
-
-theorem wcpr0_getl:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
-k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k
-u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1)
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1
-u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda
-(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C
-C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3
-(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
-(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let
-H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
-u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K
-(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
-O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r
-T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
-(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3)))))
-(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
-T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1
-u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4
-(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2
-(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5
-\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3)
-(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda
-(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
-u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f)
-u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
-x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
-(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
-u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k
-(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
-K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
-K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
-(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1)
-(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
-(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
-u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
-u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
-(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7:
-(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2)
-H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat
-f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
-T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
-u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
-e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
-(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda
-(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2)
-H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
-
-theorem wcpr0_getl_back:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
-k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
-u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
-u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k
-u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2)
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1
-u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda
-(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C
-C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4
-(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
-(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let
-H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
-u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K
-(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
-O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r
-T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
-(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t)))))
-(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
-T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3
-(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1
-(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5
-\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4)
-(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda
-(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f)
-u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
-e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
-(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k
-(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
-K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
-K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
-(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2)
-(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
-(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
-u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
-u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
-(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7:
-(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1)
-H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat
-f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
-T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
-u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
-e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
-(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda
-(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1)
-H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/wf3/fwd.ma".
-
-theorem wf3_clear_conf:
- \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall
-(c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
-\def
- \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind
-(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0
-c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u:
-T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u)
-c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e
-c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g
-c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def
-(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
-
-theorem clear_wf3_trans:
- \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall
-(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda
-(c2: C).(clear c2 d2))))))))
-\def
- \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind
-(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0
-d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2
-d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g:
-G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x
-\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T
-(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda
-(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g
-e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
-(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
-\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2))
-(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2:
-C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2:
-C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u
-w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2
-(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e
-(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e
-x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
-(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2:
-C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u)
-c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u)
-(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda
-(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
-(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
-\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void)
-(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w:
-T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind
-b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3:
-(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e
-x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C
-(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3
-g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C
-(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2
-(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void
-g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1)))))))))
-(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
-((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2:
-C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f:
-F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c
-d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda
-(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2:
-C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda
-(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C
-(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2
-d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/ty3/defs.ma".
-
-inductive wf3 (g: G): C \to (C \to Prop) \def
-| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
-| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
-T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind
-b) u) (CHead c2 (Bind b) u))))))))
-| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
-T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g
-(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
-| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
-T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/wf3/defs.ma".
-
-theorem wf3_gen_sort1:
- \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to
-(eq C x (CSort m)))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort
-m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c:
-C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda
-(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0:
-nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat
-(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n)
-\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in
-(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C
-(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
-c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4:
-(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1
-(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m)
-H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u))
-H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
-c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
-T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
-B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind
-C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
-True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O))
-(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2:
-C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C
-c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat
-f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C
-c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
-
-theorem wf3_gen_bind1:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b:
-B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2:
-C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda
-(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3
-C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2:
-C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
-False))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b:
-B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind
-b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda
-(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2:
-C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
-v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O))))
-(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
-w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or
-(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v))))
-(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w:
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void)
-(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w:
-T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C
-(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v)
-H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C
-(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1
-c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2:
-C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g
-c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2))))
-(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
-(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
-v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
-(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
-w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0
-u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1
-(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
-c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind
-b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9:
-(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v))))
-(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w:
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u)
-(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda
-(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind
-T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0:
-T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b)
-t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
-C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
-C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
-False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1
-H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b)
-v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
-c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
-\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl
-(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v)
-(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
-C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
-C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
-False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2
-(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
-c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C
-(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5)))))))))))
-(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
-(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
-v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
-(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
-w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g
-c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind
-b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
-(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v)
-H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
-(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
-(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4)
-in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
-(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0
-b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t:
-T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def
-(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10
-c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind
-b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
-c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
-\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror
-(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void)
-(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
-c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda
-(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
-c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind
-Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g
-c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2
-(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6))
-H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0
-c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T
-(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
-(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
-g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
-c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C
-(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead
-c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C
-T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
-(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
-g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
-c1 v w) \to False))))) H4))))))))) y x H0))) H)))))).
-
-theorem wf3_gen_flat1:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f:
-F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f:
-F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat
-f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y:
-C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0:
-C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m:
-nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def
-(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m))
-H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda
-(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u:
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4:
-(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C
-(CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
-_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1
-(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2:
-C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v))
-\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0
-u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u)
-(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2
-(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2:
-C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f)
-v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C
-(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead
-c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in
-C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _)
-\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead
-c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda
-(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0
-(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8)
-in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in
-H10))))) H5)) H4))))))))) y x H0))) H)))))).
-
-theorem wf3_gen_head2:
- \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k:
-K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k:
-K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda
-(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind
-b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_:
-C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
-k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k
-v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b:
-B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
-(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b:
-B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3
-g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c
-k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2
-(Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in
-(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def
-(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def
-(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda
-(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2
-(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k
-(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0
-(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B
-(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K
-(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6))
-H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
-c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
-k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u
-t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void)
-(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
-_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in
-((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
-C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow
-k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2
-(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void)
-k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0:
-C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0))))))
-H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c
-H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v))
-\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in
-(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind
-b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind
-Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12
-(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))
-Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2
-(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_:
-T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def
-(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind
-C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq
-K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda
-(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k
-v))))))))))))) x y H0))) H)))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/wf3/clear.ma".
-
-include "LambdaDelta-1/ty3/dec.ma".
-
-theorem wf3_getl_conf:
- \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall
-(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2:
-C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2)))))))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1:
-C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to
-(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g
-d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v)))
-(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1:
-C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda
-(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda
-(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v)
-(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def
-(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T
-(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
-(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3
-g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
-g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
-b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda
-(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1
-v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_:
-C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda
-(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind
-b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b)
-v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O
-(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))
-x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3:
-C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1
-c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to
-False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
-O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
-g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b)
-v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2
-(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6:
-((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind
-Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in
-(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0
-(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H:
-((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1
-(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall
-(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
-b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind
-(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1
-(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall
-(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0:
-nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0)
-(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g
-(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort
-n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda
-(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead
-d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to
-(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl
-(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3
-g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r
-k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0:
-B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r
-(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t
-b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda
-(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_:
-T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C
-(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
-C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to
-False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v)))
-(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3:
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3:
-C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t
-w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
-(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_:
-C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda
-(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind
-b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g
-x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2:
-C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2:
-C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
-b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
-n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11)))
-c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3
-(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_:
-C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3:
-C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3))
-(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda
-(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
-d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort
-O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t
-w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0:
-C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda
-(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let
-H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b)
-v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n)
-(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
-C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
-b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
-n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
-C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12
-(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda
-(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c
-(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1
-v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n
-H1)))))))))))))) c1)))) i)).
-
-theorem getl_wf3_trans:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to
-(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
-C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to
-(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2
-d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1
-d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x
-\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x
-in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2))
-(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2)))
-(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x
-d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2
-d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n:
-nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to
-(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
-C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1:
-C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall
-(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c
-c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda
-(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda
-(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C
-(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2
-d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c
-d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda
-(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k
-t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind
-(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g
-(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b:
-B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2
-H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda
-(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t)
-c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3
-g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in
-(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall
-(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c
-(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T
-(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2))
-(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2:
-C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t
-x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda
-(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9
-b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2:
-T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead
-c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void)
-(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort
-O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c
-d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C
-(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C
-(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S
-n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S
-n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2))
-(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4)))))
-k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/wf3/ty3.ma".
-
-include "LambdaDelta-1/app/defs.ma".
-
-theorem wf3_mono:
- \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall
-(c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c
-c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0
-c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0:
-(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C
-(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2
-H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2
-c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3
-c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u
-t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b)
-u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in
-(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind
-b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead
-c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u)
-c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead
-c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda
-(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4:
-C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4:
-C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2
-u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2
-x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
-(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind
-b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0
-H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind
-Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall
-(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0
-(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda
-(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b)
-u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort
-O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u
-w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4:
-C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def
-H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort
-O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3:
-C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4)
-\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g
-c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g
-(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in
-(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C
-c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4)))
-(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq
-C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4))
-(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3
-(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda
-(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_:
-T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u
-w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4
-(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
-C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O))
-c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind
-b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r
-C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort
-O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C
-(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6))))))
-H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void)
-(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w:
-T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead
-c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
-C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void)
-(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void)
-(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3
-g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda
-(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead
-c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K
-(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4))))))))))))
-(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1:
-((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u:
-T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f)
-u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y))))))))))
-c c1 H)))).
-
-theorem wf3_total:
- \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
-C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
-(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
-(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
-H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
-(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
-(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
-(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
-T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
-False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
-(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
-g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
-(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
-(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
-b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
-False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
-(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
-(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
-(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
-
-theorem ty3_shift1:
- \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall
-(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c
-t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c
-(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall
-(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0
-t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0:
-C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3
-g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2))))))))
-(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda
-(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C
-c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g
-(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda
-(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C
-(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C
-C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u)
-(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0:
-C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to
-(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8
-\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T
-(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1))
-(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x:
-T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1)
-(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2
-H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5)))))))))))))))))
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2:
-(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to
-(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u:
-T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
-B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort
-O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind
-b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
-in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
-(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k
-in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
-in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
-(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9:
-(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b
-(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in
-(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u)))
-(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12
-\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11
-(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0:
-T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O)
-(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1
-(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def
-(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall
-(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1
-t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g
-c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void)
-(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort
-O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda
-(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1)
-(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g
-c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12))))
-(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b
-H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
-(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall
-(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
-t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1
-(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead
-c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1
-(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1
-c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort
-(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in
-(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1
-(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in
-(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3
-g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
-t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def
-(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_:
-K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I
-(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u
-t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0)))
-H))).
-
-theorem wf3_idem:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
-c2 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
-c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
-nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
-c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
-(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
-c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
-(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
-((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
-c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
-(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
-c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
-
-theorem wf3_ty3:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
-u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
-u)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
-C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
-(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
-x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
-u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-1/wf3/getl.ma".
-
-theorem wf3_pr2_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
-u) \to (pr2 c2 t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
-T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2
-c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u:
-T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
-C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3
-u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def
-(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in
-(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x:
-T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0
-g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t)
-(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda
-(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7)))))
-H5)))))))))))))))))) c1 t1 t2 H))))).
-
-theorem wf3_pr3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
-u) \to (pr3 c2 t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall
-(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t
-t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1
-c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t))))))
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5:
-T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1
-c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
-(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1
-t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2
-H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
-
-theorem wf3_pc3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1
-u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda
-(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g
-c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
-(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4:
-(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf
-g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2)))))
-H3)))))))))))).
-
-theorem wf3_ty3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c:
-C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda
-(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u:
-T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall
-(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4
-t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0:
-T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4
-x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4
-c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c:
-C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g
-c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
-(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g
-c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
-(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
-C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
-C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x:
-C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d
-x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n:
-nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
-(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
-t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u
-t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
-(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
-C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x:
-C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d
-x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1:
-((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b:
-B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u)
-t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to
-(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g
-c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t
-H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
-(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
-c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
-(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
-c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
-c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda
-(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0:
-T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c
-c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
-c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/preamble.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
-include "LambdaDelta-2/T/props.ma".
-
-inline procedural "LambdaDelta-1/C/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/preamble.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
-inline procedural "LambdaDelta-1/T/dec.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/preamble.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
-inline procedural "LambdaDelta-1/T/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/asucc/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/aplus/defs.ma".
-
-include "LambdaDelta-2/next_plus/props.ma".
-
-inline procedural "LambdaDelta-1/aplus/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/A/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/aprem/defs.ma".
-
-inline procedural "LambdaDelta-1/aprem/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/aprem/fwd.ma".
-
-include "LambdaDelta-2/leq/defs.ma".
-
-inline procedural "LambdaDelta-1/aprem/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/arity/props.ma".
-
-include "LambdaDelta-2/arity/cimp.ma".
-
-include "LambdaDelta-2/aprem/props.ma".
-
-inline procedural "LambdaDelta-1/arity/aprem.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/arity/defs.ma".
-
-include "LambdaDelta-2/cimp/props.ma".
-
-inline procedural "LambdaDelta-1/arity/cimp.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/leq/defs.ma".
-
-include "LambdaDelta-2/getl/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/arity/defs.ma".
-
-include "LambdaDelta-2/leq/asucc.ma".
-
-include "LambdaDelta-2/getl/drop.ma".
-
-inline procedural "LambdaDelta-1/arity/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/arity/props.ma".
-
-include "LambdaDelta-2/drop1/fwd.ma".
-
-inline procedural "LambdaDelta-1/arity/lift1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csuba/arity.ma".
-
-include "LambdaDelta-2/pr3/defs.ma".
-
-include "LambdaDelta-2/pr1/defs.ma".
-
-include "LambdaDelta-2/wcpr0/getl.ma".
-
-include "LambdaDelta-2/pr0/fwd.ma".
-
-include "LambdaDelta-2/arity/subst0.ma".
-
-inline procedural "LambdaDelta-1/arity/pr3.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/arity/fwd.ma".
-
-inline procedural "LambdaDelta-1/arity/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/arity/props.ma".
-
-include "LambdaDelta-2/fsubst0/fwd.ma".
-
-include "LambdaDelta-2/csubst0/getl.ma".
-
-include "LambdaDelta-2/subst0/dec.ma".
-
-include "LambdaDelta-2/subst0/fwd.ma".
-
-include "LambdaDelta-2/getl/getl.ma".
-
-inline procedural "LambdaDelta-1/arity/subst0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/A/defs.ma".
-
-include "LambdaDelta-2/G/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/asucc/defs.ma".
-
-inline procedural "LambdaDelta-1/asucc/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/cimp/defs.ma".
-
-include "LambdaDelta-2/getl/getl.ma".
-
-inline procedural "LambdaDelta-1/cimp/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/clear/fwd.ma".
-
-include "LambdaDelta-2/drop/fwd.ma".
-
-inline procedural "LambdaDelta-1/clear/drop.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/clear/defs.ma".
-
-inline procedural "LambdaDelta-1/clear/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/clear/fwd.ma".
-
-inline procedural "LambdaDelta-1/clear/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
-include "LambdaDelta-2/s/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/clen/defs.ma".
-
-include "LambdaDelta-2/getl/props.ma".
-
-inline procedural "LambdaDelta-1/clen/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/cnt/defs.ma".
-
-include "LambdaDelta-2/lift/fwd.ma".
-
-inline procedural "LambdaDelta-1/cnt/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csuba/getl.ma".
-
-include "LambdaDelta-2/csuba/props.ma".
-
-include "LambdaDelta-2/arity/props.ma".
-
-include "LambdaDelta-2/csubv/getl.ma".
-
-inline procedural "LambdaDelta-1/csuba/arity.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csuba/defs.ma".
-
-include "LambdaDelta-2/clear/fwd.ma".
-
-inline procedural "LambdaDelta-1/csuba/clear.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/arity/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csuba/fwd.ma".
-
-include "LambdaDelta-2/drop/fwd.ma".
-
-inline procedural "LambdaDelta-1/csuba/drop.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csuba/defs.ma".
-
-inline procedural "LambdaDelta-1/csuba/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csuba/drop.ma".
-
-include "LambdaDelta-2/csuba/clear.ma".
-
-include "LambdaDelta-2/getl/clear.ma".
-
-inline procedural "LambdaDelta-1/csuba/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csuba/defs.ma".
-
-inline procedural "LambdaDelta-1/csuba/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/csuba.ma".
-
-inline procedural "LambdaDelta-1/csubc/arity.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubc/clear.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/defs.ma".
-
-include "LambdaDelta-2/sc3/props.ma".
-
-inline procedural "LambdaDelta-1/csubc/csuba.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sc3/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/fwd.ma".
-
-include "LambdaDelta-2/sc3/props.ma".
-
-inline procedural "LambdaDelta-1/csubc/drop.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/drop.ma".
-
-inline procedural "LambdaDelta-1/csubc/drop1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/defs.ma".
-
-inline procedural "LambdaDelta-1/csubc/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/drop.ma".
-
-include "LambdaDelta-2/csubc/clear.ma".
-
-inline procedural "LambdaDelta-1/csubc/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/defs.ma".
-
-include "LambdaDelta-2/sc3/props.ma".
-
-inline procedural "LambdaDelta-1/csubc/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst0/props.ma".
-
-include "LambdaDelta-2/csubst0/fwd.ma".
-
-include "LambdaDelta-2/clear/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubst0/clear.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/defs.ma".
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst0/fwd.ma".
-
-include "LambdaDelta-2/drop/fwd.ma".
-
-include "LambdaDelta-2/s/props.ma".
-
-inline procedural "LambdaDelta-1/csubst0/drop.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst0/defs.ma".
-
-inline procedural "LambdaDelta-1/csubst0/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst0/clear.ma".
-
-include "LambdaDelta-2/csubst0/drop.ma".
-
-include "LambdaDelta-2/getl/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubst0/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst0/defs.ma".
-
-inline procedural "LambdaDelta-1/csubst0/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst0/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst1/defs.ma".
-
-include "LambdaDelta-2/csubst0/fwd.ma".
-
-include "LambdaDelta-2/subst1/props.ma".
-
-inline procedural "LambdaDelta-1/csubst1/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst1/props.ma".
-
-include "LambdaDelta-2/csubst0/getl.ma".
-
-include "LambdaDelta-2/subst1/props.ma".
-
-include "LambdaDelta-2/drop/props.ma".
-
-inline procedural "LambdaDelta-1/csubst1/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst1/defs.ma".
-
-include "LambdaDelta-2/subst1/defs.ma".
-
-inline procedural "LambdaDelta-1/csubst1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/defs.ma".
-
-include "LambdaDelta-2/clear/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubt/clear.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/arity.ma".
-
-inline procedural "LambdaDelta-1/csubt/csuba.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/fwd.ma".
-
-include "LambdaDelta-2/drop/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubt/drop.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/defs.ma".
-
-inline procedural "LambdaDelta-1/csubt/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/clear.ma".
-
-include "LambdaDelta-2/csubt/drop.ma".
-
-include "LambdaDelta-2/getl/clear.ma".
-
-inline procedural "LambdaDelta-1/csubt/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/getl.ma".
-
-include "LambdaDelta-2/pc3/left.ma".
-
-inline procedural "LambdaDelta-1/csubt/pc3.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/defs.ma".
-
-inline procedural "LambdaDelta-1/csubt/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/pc3.ma".
-
-include "LambdaDelta-2/csubt/props.ma".
-
-inline procedural "LambdaDelta-1/csubt/ty3.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubv/defs.ma".
-
-include "LambdaDelta-2/clear/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubv/clear.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubv/props.ma".
-
-include "LambdaDelta-2/drop/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubv/drop.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubv/clear.ma".
-
-include "LambdaDelta-2/csubv/drop.ma".
-
-include "LambdaDelta-2/getl/fwd.ma".
-
-inline procedural "LambdaDelta-1/csubv/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubv/defs.ma".
-
-include "LambdaDelta-2/T/props.ma".
-
-inline procedural "LambdaDelta-1/csubv/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
-include "LambdaDelta-2/lift/defs.ma".
-
-include "LambdaDelta-2/r/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/drop/defs.ma".
-
-inline procedural "LambdaDelta-1/drop/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/drop/fwd.ma".
-
-include "LambdaDelta-2/lift/props.ma".
-
-include "LambdaDelta-2/r/props.ma".
-
-inline procedural "LambdaDelta-1/drop/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/drop/defs.ma".
-
-include "LambdaDelta-2/lift1/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/drop1/defs.ma".
-
-inline procedural "LambdaDelta-1/drop1/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/drop1/fwd.ma".
-
-include "LambdaDelta-2/getl/drop.ma".
-
-inline procedural "LambdaDelta-1/drop1/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/drop1/fwd.ma".
-
-include "LambdaDelta-2/drop/props.ma".
-
-include "LambdaDelta-2/getl/defs.ma".
-
-inline procedural "LambdaDelta-1/drop1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/A/defs.ma".
-
-include "LambdaDelta-2/G/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ex0/defs.ma".
-
-include "LambdaDelta-2/leq/defs.ma".
-
-include "LambdaDelta-2/aplus/props.ma".
-
-inline procedural "LambdaDelta-1/ex0/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ex1/defs.ma".
-
-include "LambdaDelta-2/ty3/fwd.ma".
-
-include "LambdaDelta-2/pc3/fwd.ma".
-
-include "LambdaDelta-2/nf2/pr3.ma".
-
-include "LambdaDelta-2/nf2/props.ma".
-
-include "LambdaDelta-2/arity/defs.ma".
-
-include "LambdaDelta-2/leq/props.ma".
-
-inline procedural "LambdaDelta-1/ex1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ex2/defs.ma".
-
-include "LambdaDelta-2/nf2/defs.ma".
-
-include "LambdaDelta-2/pr2/fwd.ma".
-
-include "LambdaDelta-2/arity/fwd.ma".
-
-inline procedural "LambdaDelta-1/ex2/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/flt/defs.ma".
-
-include "LambdaDelta-2/C/props.ma".
-
-inline procedural "LambdaDelta-1/flt/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubst0/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/fsubst0/defs.ma".
-
-inline procedural "LambdaDelta-1/fsubst0/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/props.ma".
-
-include "LambdaDelta-2/clear/drop.ma".
-
-inline procedural "LambdaDelta-1/getl/clear.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/props.ma".
-
-inline procedural "LambdaDelta-1/getl/dec.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/drop/defs.ma".
-
-include "LambdaDelta-2/clear/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/props.ma".
-
-include "LambdaDelta-2/clear/drop.ma".
-
-inline procedural "LambdaDelta-1/getl/drop.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/fwd.ma".
-
-include "LambdaDelta-2/clear/props.ma".
-
-include "LambdaDelta-2/flt/props.ma".
-
-inline procedural "LambdaDelta-1/getl/flt.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/defs.ma".
-
-include "LambdaDelta-2/drop/fwd.ma".
-
-include "LambdaDelta-2/clear/fwd.ma".
-
-inline procedural "LambdaDelta-1/getl/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/drop.ma".
-
-include "LambdaDelta-2/getl/clear.ma".
-
-inline procedural "LambdaDelta-1/getl/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/getl/fwd.ma".
-
-include "LambdaDelta-2/drop/props.ma".
-
-include "LambdaDelta-2/clear/props.ma".
-
-inline procedural "LambdaDelta-1/getl/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/iso/defs.ma".
-
-include "LambdaDelta-2/tlist/defs.ma".
-
-inline procedural "LambdaDelta-1/iso/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/iso/fwd.ma".
-
-inline procedural "LambdaDelta-1/iso/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/leq/props.ma".
-
-inline procedural "LambdaDelta-1/leq/asucc.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/aplus/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/leq/defs.ma".
-
-inline procedural "LambdaDelta-1/leq/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/leq/fwd.ma".
-
-include "LambdaDelta-2/aplus/props.ma".
-
-inline procedural "LambdaDelta-1/leq/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/tlist/defs.ma".
-
-include "LambdaDelta-2/s/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift/defs.ma".
-
-inline procedural "LambdaDelta-1/lift/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift/fwd.ma".
-
-include "LambdaDelta-2/s/props.ma".
-
-inline procedural "LambdaDelta-1/lift/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift/fwd.ma".
-
-include "LambdaDelta-2/tlt/props.ma".
-
-inline procedural "LambdaDelta-1/lift/tlt.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift1/defs.ma".
-
-include "LambdaDelta-2/lift/fwd.ma".
-
-inline procedural "LambdaDelta-1/lift1/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift/props.ma".
-
-include "LambdaDelta-2/drop1/defs.ma".
-
-inline procedural "LambdaDelta-1/lift1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/A/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/llt/defs.ma".
-
-include "LambdaDelta-2/leq/defs.ma".
-
-inline procedural "LambdaDelta-1/llt/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/G/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/next_plus/defs.ma".
-
-inline procedural "LambdaDelta-1/next_plus/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/nf2/fwd.ma".
-
-include "LambdaDelta-2/arity/subst0.ma".
-
-inline procedural "LambdaDelta-1/nf2/arity.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/nf2/defs.ma".
-
-include "LambdaDelta-2/pr2/clen.ma".
-
-include "LambdaDelta-2/pr2/fwd.ma".
-
-include "LambdaDelta-2/pr0/dec.ma".
-
-include "LambdaDelta-2/C/props.ma".
-
-inline procedural "LambdaDelta-1/nf2/dec.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr2/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/nf2/defs.ma".
-
-include "LambdaDelta-2/pr2/clen.ma".
-
-include "LambdaDelta-2/subst0/dec.ma".
-
-include "LambdaDelta-2/T/props.ma".
-
-inline procedural "LambdaDelta-1/nf2/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/nf2/pr3.ma".
-
-include "LambdaDelta-2/pr3/fwd.ma".
-
-include "LambdaDelta-2/iso/props.ma".
-
-inline procedural "LambdaDelta-1/nf2/iso.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/nf2/props.ma".
-
-include "LambdaDelta-2/drop1/fwd.ma".
-
-inline procedural "LambdaDelta-1/nf2/lift1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/nf2/defs.ma".
-
-include "LambdaDelta-2/pr3/pr3.ma".
-
-inline procedural "LambdaDelta-1/nf2/pr3.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/nf2/defs.ma".
-
-include "LambdaDelta-2/pr2/fwd.ma".
-
-inline procedural "LambdaDelta-1/nf2/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr1/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc1/defs.ma".
-
-include "LambdaDelta-2/pr1/pr1.ma".
-
-inline procedural "LambdaDelta-1/pc1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/arity_props.ma".
-
-include "LambdaDelta-2/nf2/fwd.ma".
-
-inline procedural "LambdaDelta-1/pc3/dec.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/left.ma".
-
-include "LambdaDelta-2/fsubst0/defs.ma".
-
-include "LambdaDelta-2/csubst0/getl.ma".
-
-inline procedural "LambdaDelta-1/pc3/fsubst0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/props.ma".
-
-include "LambdaDelta-2/pr3/fwd.ma".
-
-inline procedural "LambdaDelta-1/pc3/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/props.ma".
-
-inline procedural "LambdaDelta-1/pc3/left.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/defs.ma".
-
-include "LambdaDelta-2/nf2/pr3.ma".
-
-inline procedural "LambdaDelta-1/pc3/nf2.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/defs.ma".
-
-include "LambdaDelta-2/pc1/defs.ma".
-
-include "LambdaDelta-2/pr3/pr1.ma".
-
-inline procedural "LambdaDelta-1/pc3/pc1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/defs.ma".
-
-include "LambdaDelta-2/pr3/pr3.ma".
-
-inline procedural "LambdaDelta-1/pc3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/props.ma".
-
-include "LambdaDelta-2/pr3/subst1.ma".
-
-inline procedural "LambdaDelta-1/pc3/subst1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/props.ma".
-
-include "LambdaDelta-2/wcpr0/getl.ma".
-
-inline procedural "LambdaDelta-1/pc3/wcpr0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/fwd.ma".
-
-include "LambdaDelta-2/subst0/dec.ma".
-
-include "LambdaDelta-2/T/dec.ma".
-
-include "LambdaDelta-2/T/props.ma".
-
-inline procedural "LambdaDelta-1/pr0/dec.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/props.ma".
-
-inline procedural "LambdaDelta-1/pr0/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/fwd.ma".
-
-include "LambdaDelta-2/lift/tlt.ma".
-
-inline procedural "LambdaDelta-1/pr0/pr0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/defs.ma".
-
-include "LambdaDelta-2/subst0/subst0.ma".
-
-inline procedural "LambdaDelta-1/pr0/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/props.ma".
-
-include "LambdaDelta-2/subst1/defs.ma".
-
-inline procedural "LambdaDelta-1/pr0/subst1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr1/props.ma".
-
-include "LambdaDelta-2/pr0/pr0.ma".
-
-inline procedural "LambdaDelta-1/pr1/pr1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr1/defs.ma".
-
-include "LambdaDelta-2/pr0/subst1.ma".
-
-include "LambdaDelta-2/subst1/props.ma".
-
-include "LambdaDelta-2/T/props.ma".
-
-inline procedural "LambdaDelta-1/pr1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr2/props.ma".
-
-include "LambdaDelta-2/clen/getl.ma".
-
-inline procedural "LambdaDelta-1/pr2/clen.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/defs.ma".
-
-include "LambdaDelta-2/getl/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr2/defs.ma".
-
-include "LambdaDelta-2/pr0/fwd.ma".
-
-include "LambdaDelta-2/getl/drop.ma".
-
-include "LambdaDelta-2/getl/clear.ma".
-
-inline procedural "LambdaDelta-1/pr2/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr2/defs.ma".
-
-include "LambdaDelta-2/pr0/pr0.ma".
-
-include "LambdaDelta-2/getl/props.ma".
-
-inline procedural "LambdaDelta-1/pr2/pr2.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr2/defs.ma".
-
-include "LambdaDelta-2/pr0/props.ma".
-
-include "LambdaDelta-2/getl/drop.ma".
-
-include "LambdaDelta-2/getl/clear.ma".
-
-inline procedural "LambdaDelta-1/pr2/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr2/defs.ma".
-
-include "LambdaDelta-2/pr0/subst1.ma".
-
-include "LambdaDelta-2/pr0/fwd.ma".
-
-include "LambdaDelta-2/csubst1/getl.ma".
-
-include "LambdaDelta-2/csubst1/fwd.ma".
-
-include "LambdaDelta-2/subst1/subst1.ma".
-
-include "LambdaDelta-2/getl/drop.ma".
-
-inline procedural "LambdaDelta-1/pr2/subst1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr2/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/props.ma".
-
-include "LambdaDelta-2/pr2/fwd.ma".
-
-inline procedural "LambdaDelta-1/pr3/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/fwd.ma".
-
-include "LambdaDelta-2/iso/props.ma".
-
-include "LambdaDelta-2/tlist/props.ma".
-
-inline procedural "LambdaDelta-1/pr3/iso.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/defs.ma".
-
-include "LambdaDelta-2/pr1/defs.ma".
-
-inline procedural "LambdaDelta-1/pr3/pr1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/props.ma".
-
-include "LambdaDelta-2/pr2/pr2.ma".
-
-inline procedural "LambdaDelta-1/pr3/pr3.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/pr1.ma".
-
-include "LambdaDelta-2/pr2/props.ma".
-
-include "LambdaDelta-2/pr1/props.ma".
-
-inline procedural "LambdaDelta-1/pr3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/defs.ma".
-
-include "LambdaDelta-2/pr2/subst1.ma".
-
-inline procedural "LambdaDelta-1/pr3/subst1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/props.ma".
-
-include "LambdaDelta-2/wcpr0/getl.ma".
-
-inline procedural "LambdaDelta-1/pr3/wcpr0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Base-2/theory.ma".
-include "LambdaDelta-1/definitions.ma".
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/r/defs.ma".
-
-include "LambdaDelta-2/s/defs.ma".
-
-inline procedural "LambdaDelta-1/r/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/s/defs.ma".
-
-inline procedural "LambdaDelta-1/s/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubc/arity.ma".
-
-include "LambdaDelta-2/csubc/getl.ma".
-
-include "LambdaDelta-2/csubc/drop1.ma".
-
-include "LambdaDelta-2/csubc/props.ma".
-
-inline procedural "LambdaDelta-1/sc3/arity.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sn3/defs.ma".
-
-include "LambdaDelta-2/arity/defs.ma".
-
-include "LambdaDelta-2/drop1/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sc3/defs.ma".
-
-include "LambdaDelta-2/sn3/lift1.ma".
-
-include "LambdaDelta-2/nf2/lift1.ma".
-
-include "LambdaDelta-2/csuba/arity.ma".
-
-include "LambdaDelta-2/arity/lift1.ma".
-
-include "LambdaDelta-2/arity/aprem.ma".
-
-include "LambdaDelta-2/llt/props.ma".
-
-include "LambdaDelta-2/drop1/getl.ma".
-
-include "LambdaDelta-2/drop1/props.ma".
-
-include "LambdaDelta-2/lift1/props.ma".
-
-inline procedural "LambdaDelta-1/sc3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr3/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sn3/defs.ma".
-
-include "LambdaDelta-2/pr3/props.ma".
-
-inline procedural "LambdaDelta-1/sn3/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sn3/props.ma".
-
-include "LambdaDelta-2/drop1/fwd.ma".
-
-include "LambdaDelta-2/lift1/fwd.ma".
-
-inline procedural "LambdaDelta-1/sn3/lift1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sn3/defs.ma".
-
-include "LambdaDelta-2/nf2/dec.ma".
-
-include "LambdaDelta-2/nf2/pr3.ma".
-
-inline procedural "LambdaDelta-1/sn3/nf2.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sn3/nf2.ma".
-
-include "LambdaDelta-2/sn3/fwd.ma".
-
-include "LambdaDelta-2/nf2/iso.ma".
-
-include "LambdaDelta-2/pr3/iso.ma".
-
-inline procedural "LambdaDelta-1/sn3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/G/defs.ma".
-
-include "LambdaDelta-2/getl/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sty0/defs.ma".
-
-inline procedural "LambdaDelta-1/sty0/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sty0/defs.ma".
-
-include "LambdaDelta-2/getl/drop.ma".
-
-inline procedural "LambdaDelta-1/sty0/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sty1/props.ma".
-
-include "LambdaDelta-2/cnt/props.ma".
-
-inline procedural "LambdaDelta-1/sty1/cnt.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sty0/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/sty1/defs.ma".
-
-include "LambdaDelta-2/sty0/props.ma".
-
-inline procedural "LambdaDelta-1/sty1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst/defs.ma".
-
-inline procedural "LambdaDelta-1/subst/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst/fwd.ma".
-
-include "LambdaDelta-2/subst0/defs.ma".
-
-include "LambdaDelta-2/lift/props.ma".
-
-inline procedural "LambdaDelta-1/subst/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/defs.ma".
-
-include "LambdaDelta-2/lift/props.ma".
-
-inline procedural "LambdaDelta-1/subst0/dec.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/lift/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/defs.ma".
-
-include "LambdaDelta-2/lift/props.ma".
-
-inline procedural "LambdaDelta-1/subst0/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/fwd.ma".
-
-inline procedural "LambdaDelta-1/subst0/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/props.ma".
-
-inline procedural "LambdaDelta-1/subst0/subst0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/defs.ma".
-
-include "LambdaDelta-2/lift/props.ma".
-
-include "LambdaDelta-2/lift/tlt.ma".
-
-inline procedural "LambdaDelta-1/subst0/tlt.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst1/defs.ma".
-
-include "LambdaDelta-2/subst0/props.ma".
-
-inline procedural "LambdaDelta-1/subst1/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst1/defs.ma".
-
-include "LambdaDelta-2/subst0/props.ma".
-
-inline procedural "LambdaDelta-1/subst1/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst1/fwd.ma".
-
-include "LambdaDelta-2/subst0/subst0.ma".
-
-inline procedural "LambdaDelta-1/subst1/subst1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/subst0/tlt.ma".
-
-include "LambdaDelta-2/subst/props.ma".
-
-include "LambdaDelta-2/sty1/cnt.ma".
-
-include "LambdaDelta-2/ex0/props.ma".
-
-include "LambdaDelta-2/wcpr0/fwd.ma".
-
-include "LambdaDelta-2/pr3/wcpr0.ma".
-
-include "LambdaDelta-2/ex2/props.ma".
-
-include "LambdaDelta-2/ex1/props.ma".
-
-include "LambdaDelta-2/ty3/sty0.ma".
-
-include "LambdaDelta-2/csubt/csuba.ma".
-
-include "LambdaDelta-2/ty3/fwd_nf2.ma".
-
-include "LambdaDelta-2/ty3/nf2.ma".
-
-include "LambdaDelta-2/wf3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/tlist/defs.ma".
-
-inline procedural "LambdaDelta-1/tlist/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/T/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/tlt/defs.ma".
-
-inline procedural "LambdaDelta-1/tlt/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/pr3_props.ma".
-
-include "LambdaDelta-2/arity/pr3.ma".
-
-include "LambdaDelta-2/asucc/fwd.ma".
-
-inline procedural "LambdaDelta-1/ty3/arity.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/arity.ma".
-
-include "LambdaDelta-2/sc3/arity.ma".
-
-inline procedural "LambdaDelta-1/ty3/arity_props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pc3/dec.ma".
-
-include "LambdaDelta-2/getl/flt.ma".
-
-include "LambdaDelta-2/getl/dec.ma".
-
-inline procedural "LambdaDelta-1/ty3/dec.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/G/defs.ma".
-
-include "LambdaDelta-2/pc3/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/props.ma".
-
-include "LambdaDelta-2/pc3/fsubst0.ma".
-
-include "LambdaDelta-2/getl/getl.ma".
-
-inline procedural "LambdaDelta-1/ty3/fsubst0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/defs.ma".
-
-include "LambdaDelta-2/pc3/props.ma".
-
-inline procedural "LambdaDelta-1/ty3/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/arity_props.ma".
-
-include "LambdaDelta-2/pc3/nf2.ma".
-
-include "LambdaDelta-2/nf2/fwd.ma".
-
-inline procedural "LambdaDelta-1/ty3/fwd_nf2.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/arity.ma".
-
-include "LambdaDelta-2/pc3/nf2.ma".
-
-include "LambdaDelta-2/nf2/arity.ma".
-
-inline procedural "LambdaDelta-1/ty3/nf2.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/csubt/ty3.ma".
-
-include "LambdaDelta-2/ty3/subst1.ma".
-
-include "LambdaDelta-2/ty3/fsubst0.ma".
-
-include "LambdaDelta-2/pc3/pc1.ma".
-
-include "LambdaDelta-2/pc3/wcpr0.ma".
-
-include "LambdaDelta-2/pc1/props.ma".
-
-inline procedural "LambdaDelta-1/ty3/pr3.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/pr3.ma".
-
-inline procedural "LambdaDelta-1/ty3/pr3_props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/fwd.ma".
-
-include "LambdaDelta-2/pc3/fwd.ma".
-
-inline procedural "LambdaDelta-1/ty3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/pr3_props.ma".
-
-include "LambdaDelta-2/sty0/fwd.ma".
-
-inline procedural "LambdaDelta-1/ty3/sty0.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/props.ma".
-
-include "LambdaDelta-2/pc3/subst1.ma".
-
-include "LambdaDelta-2/getl/getl.ma".
-
-inline procedural "LambdaDelta-1/ty3/subst1.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/pr0/defs.ma".
-
-include "LambdaDelta-2/C/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/wcpr0/defs.ma".
-
-inline procedural "LambdaDelta-1/wcpr0/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/wcpr0/defs.ma".
-
-include "LambdaDelta-2/getl/props.ma".
-
-inline procedural "LambdaDelta-1/wcpr0/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/wf3/fwd.ma".
-
-inline procedural "LambdaDelta-1/wf3/clear.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/ty3/defs.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/wf3/defs.ma".
-
-inline procedural "LambdaDelta-1/wf3/fwd.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/wf3/clear.ma".
-
-include "LambdaDelta-2/ty3/dec.ma".
-
-inline procedural "LambdaDelta-1/wf3/getl.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/wf3/ty3.ma".
-
-include "LambdaDelta-2/app/defs.ma".
-
-inline procedural "LambdaDelta-1/wf3/props.ma".
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "LambdaDelta-2/wf3/getl.ma".
-
-inline procedural "LambdaDelta-1/wf3/ty3.ma".
-
\lambda (A: Set).(\lambda (B: Set).(\lambda (f: ((A \to B))).(\lambda (x:
A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x (\lambda (a: A).(eq B
(f x) (f a))) (refl_equal B (f x)) y H)))))).
+(* COMMENTS
+Initial nodes: 51
+END *)
theorem f_equal2:
\forall (A1: Set).(\forall (A2: Set).(\forall (B: Set).(\forall (f: ((A1 \to
x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 y2)).(eq_ind
A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B (f x1 x2)) y2
H0)) y1 H))))))))).
+(* COMMENTS
+Initial nodes: 109
+END *)
theorem f_equal3:
\forall (A1: Set).(\forall (A2: Set).(\forall (A3: Set).(\forall (B:
x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 x3 (\lambda (a:
A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 x3)) y3 H1)) y2
H0)) y1 H)))))))))))).
+(* COMMENTS
+Initial nodes: 183
+END *)
theorem sym_eq:
\forall (A: Set).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y
\def
\lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x
y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))).
+(* COMMENTS
+Initial nodes: 39
+END *)
theorem eq_ind_r:
\forall (A: Set).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to
(P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) in
eq return (\lambda (a: A).(\lambda (_: (eq ? ? a)).(P a))) with [refl_equal
\Rightarrow H])))))).
+(* COMMENTS
+Initial nodes: 38
+END *)
theorem trans_eq:
\forall (A: Set).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A x y)
\lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (z: A).(\lambda
(H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda (a: A).(eq A x
a)) H z H0)))))).
+(* COMMENTS
+Initial nodes: 45
+END *)
theorem sym_not_eq:
\forall (A: Set).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to (not
\lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq A x
y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a y))
(refl_equal A y) x h2)))))).
+(* COMMENTS
+Initial nodes: 51
+END *)
theorem nat_double_ind:
\forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to
nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind
(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R
(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))).
+(* COMMENTS
+Initial nodes: 111
+END *)
theorem eq_add_S:
\forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S
m))).(f_equal nat nat pred (S n) (S m) H))).
+(* COMMENTS
+Initial nodes: 33
+END *)
theorem O_S:
\forall (n: nat).(not (eq nat O (S n)))
\def
\lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda
(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))).
+(* COMMENTS
+Initial nodes: 41
+END *)
theorem not_eq_S:
\forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda
(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))).
+(* COMMENTS
+Initial nodes: 35
+END *)
theorem pred_Sn:
\forall (m: nat).(eq nat m (pred (S m)))
\def
\lambda (m: nat).(refl_equal nat (pred (S m))).
+(* COMMENTS
+Initial nodes: 11
+END *)
theorem S_pred:
\forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S
m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0
(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))).
+(* COMMENTS
+Initial nodes: 79
+END *)
theorem le_trans:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p)
m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H
(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n
m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 57
+END *)
theorem le_trans_S:
\forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S
n) m (le_S n n (le_n n)) H))).
+(* COMMENTS
+Initial nodes: 33
+END *)
theorem le_n_S:
\forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le
n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))).
+(* COMMENTS
+Initial nodes: 65
+END *)
theorem le_O_n:
\forall (n: nat).(le O n)
\def
\lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda
(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n).
+(* COMMENTS
+Initial nodes: 33
+END *)
theorem le_S_n:
\forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0:
nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S
n m0 H0)))) (S m) H))).
+(* COMMENTS
+Initial nodes: 69
+END *)
theorem le_Sn_O:
\forall (n: nat).(not (le (S n) O))
\lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0:
nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_:
(IsSucc m)).I))) O H)).
+(* COMMENTS
+Initial nodes: 43
+END *)
theorem le_Sn_n:
\forall (n: nat).(not (le (S n) n))
\lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O
O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le
(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n).
+(* COMMENTS
+Initial nodes: 57
+END *)
theorem le_antisym:
\forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n
\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S
m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0)
in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))).
+(* COMMENTS
+Initial nodes: 119
+END *)
theorem le_n_O_eq:
\forall (n: nat).((le n O) \to (eq nat O n))
\def
\lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)).
+(* COMMENTS
+Initial nodes: 19
+END *)
theorem le_elim_rel:
\forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to
n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_:
(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0
H1)))))) m Le))))) n)))).
+(* COMMENTS
+Initial nodes: 181
+END *)
theorem lt_n_n:
\forall (n: nat).(not (lt n n))
\def
le_Sn_n.
+(* COMMENTS
+Initial nodes: 1
+END *)
theorem lt_n_S:
\forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m
H))).
+(* COMMENTS
+Initial nodes: 19
+END *)
theorem lt_n_Sn:
\forall (n: nat).(lt n (S n))
\def
\lambda (n: nat).(le_n (S n)).
+(* COMMENTS
+Initial nodes: 7
+END *)
theorem lt_S_n:
\forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S
n) m H))).
+(* COMMENTS
+Initial nodes: 23
+END *)
theorem lt_n_O:
\forall (n: nat).(not (lt n O))
\def
le_Sn_O.
+(* COMMENTS
+Initial nodes: 1
+END *)
theorem lt_trans:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p)
m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S
(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt
n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 71
+END *)
theorem lt_O_Sn:
\forall (n: nat).(lt O (S n))
\def
\lambda (n: nat).(le_n_S O n (le_O_n n)).
+(* COMMENTS
+Initial nodes: 11
+END *)
theorem lt_le_S:
\forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
\def
\lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
+(* COMMENTS
+Initial nodes: 11
+END *)
theorem le_not_lt:
\forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n
m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle
(le_trans_S (S m0) n H1)))))) m H))).
+(* COMMENTS
+Initial nodes: 67
+END *)
theorem le_lt_n_Sm:
\forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
+(* COMMENTS
+Initial nodes: 17
+END *)
theorem le_lt_trans:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p)
m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0))
(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle:
(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 69
+END *)
theorem lt_le_trans:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p)
m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H
(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S
(S n) m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 59
+END *)
theorem lt_le_weak:
\forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m
H))).
+(* COMMENTS
+Initial nodes: 17
+END *)
theorem lt_n_Sm_le:
\forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m
H))).
+(* COMMENTS
+Initial nodes: 19
+END *)
theorem le_lt_or_eq:
\forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_:
(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0))
(le_n_S n m0 H0))))) m H))).
+(* COMMENTS
+Initial nodes: 109
+END *)
theorem le_or_lt:
\forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S
n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S
m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)).
+(* COMMENTS
+Initial nodes: 209
+END *)
theorem plus_n_O:
\forall (n: nat).(eq nat n (plus n O))
\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O)))
(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0
O))).(f_equal nat nat S n0 (plus n0 O) H))) n).
+(* COMMENTS
+Initial nodes: 57
+END *)
theorem plus_n_Sm:
\forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0:
nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat
S (S (plus n0 n)) (plus n0 (S n)) H))) m)).
+(* COMMENTS
+Initial nodes: 85
+END *)
theorem plus_sym:
\forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S
(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y))
(plus_n_Sm m y)))) n)).
+(* COMMENTS
+Initial nodes: 111
+END *)
theorem plus_Snm_nSm:
\forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda
(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n
(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))).
+(* COMMENTS
+Initial nodes: 99
+END *)
theorem plus_assoc_l:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m
(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p))
(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0
m) p) H))) n))).
+(* COMMENTS
+Initial nodes: 101
+END *)
theorem plus_assoc_r:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n
(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))).
+(* COMMENTS
+Initial nodes: 37
+END *)
theorem simpl_plus_l:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m)
nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn
(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p)
(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n).
+(* COMMENTS
+Initial nodes: 161
+END *)
theorem minus_n_O:
\forall (n: nat).(eq nat n (minus n O))
\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O)))
(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0
O))).(refl_equal nat (S n0)))) n).
+(* COMMENTS
+Initial nodes: 47
+END *)
theorem minus_n_n:
\forall (n: nat).(eq nat O (minus n n))
\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0)))
(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0
n0))).IHn)) n).
+(* COMMENTS
+Initial nodes: 41
+END *)
theorem minus_Sn_m:
\forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m))
(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_:
(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow
(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))).
+(* COMMENTS
+Initial nodes: 111
+END *)
theorem plus_minus:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p))
(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq
nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H
(eq_add_S m0 (plus n0 p) H0)))))) m n))).
+(* COMMENTS
+Initial nodes: 199
+END *)
theorem minus_plus:
\forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
\def
\lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n)
(plus_minus (plus n m) n m (refl_equal nat (plus n m))))).
+(* COMMENTS
+Initial nodes: 41
+END *)
theorem le_pred_n:
\forall (n: nat).(le (pred n) n)
\lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O)
(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0
(le_n n0)))) n).
+(* COMMENTS
+Initial nodes: 43
+END *)
theorem le_plus_l:
\forall (n: nat).(\forall (m: nat).(le n (plus n m)))
n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn:
((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus
n0 m) (IHn m))))) n).
+(* COMMENTS
+Initial nodes: 55
+END *)
theorem le_plus_r:
\forall (n: nat).(\forall (m: nat).(le m (plus n m)))
\lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus
n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m
(plus n0 m) H))) n)).
+(* COMMENTS
+Initial nodes: 47
+END *)
theorem simpl_le_plus_l:
\forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p
(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda
(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus
p0 m) H))))))) p).
+(* COMMENTS
+Initial nodes: 113
+END *)
theorem le_plus_trans:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n
m)).(le_trans n m (plus m p) H (le_plus_l m p))))).
+(* COMMENTS
+Initial nodes: 31
+END *)
theorem le_reg_l:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus
(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0
m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H)))))
p))).
+(* COMMENTS
+Initial nodes: 85
+END *)
theorem le_plus_plus:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le
nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0:
nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S
(plus n p) (plus m0 q) H2)))) m H)))))).
+(* COMMENTS
+Initial nodes: 91
+END *)
theorem le_plus_minus:
\forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m
(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda
(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat
S q (plus p (minus q p)) H0))))) n m Le))).
+(* COMMENTS
+Initial nodes: 91
+END *)
theorem le_plus_minus_r:
\forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m
(plus n (minus m n)) (le_plus_minus n m H)))).
+(* COMMENTS
+Initial nodes: 33
+END *)
theorem simpl_lt_plus_l:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p
(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n
m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S
(plus p0 n)) (plus p0 m) H))))) p))).
+(* COMMENTS
+Initial nodes: 99
+END *)
theorem lt_reg_l:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus
(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0
m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H)))))
p))).
+(* COMMENTS
+Initial nodes: 85
+END *)
theorem lt_reg_r:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus
nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt
(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m
p)) (plus n p) (plus_sym n p))))).
+(* COMMENTS
+Initial nodes: 129
+END *)
theorem le_lt_plus_plus:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le
nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n
(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0)
(plus (S n) p) (plus_Snm_nSm n p))))))).
+(* COMMENTS
+Initial nodes: 75
+END *)
theorem lt_le_plus_plus:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt
\lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m
p q H H0)))))).
+(* COMMENTS
+Initial nodes: 37
+END *)
theorem lt_plus_plus:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt
\lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q
H (lt_le_weak p q H0))))))).
+(* COMMENTS
+Initial nodes: 39
+END *)
theorem well_founded_ltof:
\forall (A: Set).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb
(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a
(le_n (S (f a))))))).
+(* COMMENTS
+Initial nodes: 189
+END *)
theorem lt_wf:
well_founded nat lt
\def
well_founded_ltof nat (\lambda (m: nat).m).
+(* COMMENTS
+Initial nodes: 7
+END *)
theorem lt_wf_ind:
\forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n:
(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y:
nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y
x) \to (P y))))).(H x H1)))) p (lt_wf p)))).
+(* COMMENTS
+Initial nodes: 77
+END *)
projects / helm.git / commitdiff